%grobGeom010926.tex
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%%%%%%% Gr"obner geometry of Schubert polynomials %%%%%%%
%%%%%%% %%%%%%%
%%%%%%% Allen Knutson and Ezra Miller %%%%%%%
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\begin{document}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\title{Gr\"obner geometry of Schubert polynomials}
\author{Allen Knutson\thanks{AK was partly supported by the Clay
Mathematics Institute, Sloan Foundation, and NSF.}
\ and Ezra Miller\thanks{EM was supported by the Sloan Foundation and
NSF.}}
%\email{allenk@math.berkeley.edu}
%\address{Mathematics Department\\ UC Berkeley\\ Berkeley, California}
%\author{Ezra Miller}
%\email{ezra@math.mit.edu}
%\address{Mathematics Department\\ MIT\\ Cambridge, Massachusetts}
\date{29 December 2001}
\maketitle
\begin{abstract}
\noindent
Schubert polynomials, which a priori represent cohomology classes of
Schubert varieties in the flag manifold, also represent torus-equivariant
cohomology classes of certain determinantal loci in the vector space of
$n \times n$ complex matrices. Our central result is that the minors
defining these ``matrix Schubert varieties'' are Gr\"obner bases for any
antidiagonal term order. The Schubert polynomials are therefore positive
sums of monomials, each monomial representing the torus-equivariant
cohomology class of a component (a scheme-theoretically reduced
coordinate subspace) in the limit of the resulting Gr\"obner degeneration.
Interpreting the Hilbert series of the flat limit in equivariant
$K$-theory, another corollary of the proof is that Grothendieck
polynomials represent the classes of Schubert varieties in $K$-theory of
the flag manifold.
An inductive procedure for listing the limit coordinate subspaces is
provided by the proof of the Gr\"obner basis property, bypassing what has
come to be known as Kohnert's conjecture \cite{NoteSchubPoly}. The
coordinate subspaces, which are the facets of a simplicial complex, are
in an obvious bijection with the rc-graphs of Fomin and Kirillov
\cite{FKyangBax}. Thus our positive formula for Schubert polynomials
agrees with (and provides a geometric proof of) the combinatorial formula
of Billey-Jockusch-Stanley \cite{BJS}. Moreover, we shell this complex
(as one of a new class of vertex-decomposable complexes we introduce),
which shows that the initial ideal of the minors is a Cohen--Macaulay
Stanley--Reisner ideal. This provides a new proof that Schubert
varieties are Cohen--Macaulay.
The multidegree of any finitely generated multigraded module, defined
here based on torus-equivariant cohomology classes, generalizes the usual
$\ZZ$-graded degree to finer gradings. Part of the Gr\"obner basis
theorem includes formulae for the multidegrees and Hilbert series of
determinantal ideals in terms of Schubert and Grothendieck polynomials.
In the special case of vexillary determinantal loci, which include all
one-sided ladder determinantal varieties, the multidegree formulae are
themselves determinantal, and our new antidiagonal Gr\"obner basis
statement contrasts with known diagonal \hbox{Gr\"obner basis
statements}.
Interpreting the Schubert polynomials as equivariant cohomology classes
on matrices gives a topological reason (see also \cite{FRthomPoly})
why Schubert polynomials are the characteristic classes for
degeneracy loci \cite{FulDegLoc}: the mixing space construction of Borel
that computes this equivariant cohomology is identified as the
classifying space for maps between flagged vector bundles.
%\vskip 1ex
%\noindent
%{{\it AMS Classification:} ; }
\end{abstract}
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{}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\setcounter{tocdepth}{2}
\tableofcontents
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\section{Introduction}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:intro}
\subsection{Summary of main results}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\comment{This subsection is NOT complete.}
\subsubsection{Motivation}%Positive formulae via multidegrees}
Combinatorialists have recognized for some time the intrinsic interest of
the Schubert polynomials $\SS_w$ of Lascoux and Sch\"utzenberger indexed
by permutations $w \in S_n$, and are therefore producing a wealth of
interpretations for their coefficients; see
\cite{BergeronCombConstSchub}, \cite[Appendix to Chapter IV, by N.\
Bergeron]{NoteSchubPoly}, \cite{BJS}, \cite{FKyangBax},
\cite{FSnilCoxeter}, \cite{Kohnert}, and \cite{Winkel}. Geometers, on
the other hand, who take Schubert {\em classes} for granted, generally
remain less convinced of the naturality of Schubert {\em polynomials},
even though they arise in certain universal geometric contexts
\cite{FulDegLoc}, and there are geometric proofs of positivity for their
coefficients \cite{BSskewSchub, KoganThesis}.
Our primary motivation for undertaking this project was to provide a
geometric context in which both (i)~polynomial representatives for the
Schubert classes $[X_w]$ in $H^*(\FL)$ are uniquely singled out, with no
choices other than a Borel subgroup of $\gln$;
% (Theorem~\ref{notthm:oracle});
and (ii)~it is geometrically obvious that these representatives have
nonnegative coefficients.
% (Theorem~\ref{thm:positive}).
The fact that these polynomials turn out to be the Schubert polynomials
is a testament to their naturality. Here is the kernel of our idea.
We first undertake to replace topology on the flag manifold with
multigraded commutative algebra, as follows. The ordinary cohomology
$H^*(\FL)$ of the manifold of flags in $\CC^n$ is naturally isomorphic to
the $B$-equivariant cohomology $H^*_B(\gln)$ for the Borel subgroup $B$
inside $\gln\CC$ whose coset space is~$\FL$. Furthermore, $H^*(\FL)$ is
naturally surjected onto by the $B$-equivariant cohomology $H^*_B(\mn)
\cong \ZZ[x_1,\ldots,x_n]$ for the $n \times n$ matrices $\mn$
over~$\CC$. We therefore have a geometric explanation for Borel's
presentation of $H^*(\FL)$ as a quotient of a polynomial ring
(Lemma~\ref{lemma:borel}). In this presentation, the closure $\ol X_w
\subseteq \mn$ of the preimage of the Schubert variety $X_w$ in $\gln\CC$
yields a $B$-equivariant cohomology class $[\ol X_w]_B$ on $\mn$,
well-defined as a polynomial and mapping to $[X_w]$. Making use of the
standard equality $H^*_B(\mn) = H^*_T(\mn)$ for the maximal torus $T
\subset B$ allows us to work instead with the corresponding class $[\ol
X_w]_T$ in the $T$-equivariant cohomology of~$\mn$.
Let $W = \ZZ^n$ denote the weight lattice (character group) of our
torus~$T$. Working now with a $T$-action on a vector space~$V = \CC^m$
(thought of as~$\mn$, with $m = n^2$), we can at this point replace the
technology of equivariant cohomology by the more algebraically appealing
notion of \bem{multidegree} for subschemes of~$V$ stable under the action
of~$T$. More algebraically, the coordinate ring of~$V$, the polynomial
ring $\CC[\zz]$ in $m$ variables $z_1,\ldots,z_m$, becomes $\ZZ^n$-graded
by virtue of the torus action, with the weight of $z_i$ being the weight
of~$T$ on the $i^\th$ coordinate subspace of~$V$. The multidegree is
then an invariant associated to each graded ideal in $\CC[\zz]$, taking
values in the polynomial ring $\ZZ[x_1,\ldots,x_n] = \sym^\spot_\ZZ(W)$.
We define the multidegree for a $T$-stable subscheme of~$V$ as follows:
\begin{itemize}
\item
If $V' \subseteq V$ is a (reduced) linear subspace, then its multidegree
$[V']$ is the product of the weights of $T$'s action on $V/V'$. In
particular, when $V'$ is the zero set of $\$,
its multidegree equals the product $\,{\rm weight}(z_{i_1}) \cdots {\rm
weight}(z_{i_r})$.
\item
If $X$ is a (nonreduced) scheme supported on a subspace $V'$, and
generically of length $l$, then $[X] = l [V']$.
\item
If $X = \bigcup X_i$ is a union of irreducible components~$X_i$, then
$[X] = \sum_i [X_i]$ where the sum runs over the components of top
dimension.
\item
If there is a $T$-equivariant flat family connecting two subschemes $X$
and $Y$, then they have the same multidegree. In particular, if $X$ is
the zero scheme of~$I$ and $Y$ is the zero scheme of the initial ideal
${\sf in}(I)$ for some term order, then $X$ and $Y$ have equal
multidegrees.
\end{itemize}
Part of what we show is that this gives a well-defined multidegree to any
$T$-invariant subscheme. In the case of affine cones, where $T = \CC^*$
and $\CC[\zz]$ carries its usual $\ZZ$-grading, this reduces to the
ordinary notion of degree of the corresponding projective variety. As
far as we know, this material is new, though nonetheless quite basic,
being just an algebraic reformulation of the $T$-equivariant cohomology
of affine space, or equivalently the Chow ring
\cite{TotChowRing,EGequivInt}. Note that it works just as well over any
field~$\kk$, not necessarily algebraically closed or of characteristic
zero.
Returning to the case of~$\mn$, once we relax the condition of stability
under the action of $B$ to just $T$, we can degenerate the matrix
Schubert variety $\ol X_w$ to a union $\LL_w$ of linear subspaces of
$\mn$ by finding a Gr\"obner basis for the ideal of $\ol X_w$. The
resulting flat deformation is $T$-equivariant and the limit is
$T$-stable, so the multidegrees $[\ol X_w]_T$ and $[\LL_w]_T$ are equal.
The class of a single $T$-invariant subspace is a monomial, so the
positivity of coefficients in $[\ol X_w]_T$
% (Theorem~\ref{thm:positive})
comes from the identity $[\LL_w]_T = \sum_{L \in \LL_w} [L]_T$ reflecting
the additivity of multidegrees on irreducible components.
\begin{example}
Let $w=2143 \in S_4$ (all our permutations will be written in one-line,
not cycle, notation). The matrix Schubert variety $\ol X_w$ is then the
set of $4\times 4$ matrices $Z = (z_{ij})$ whose upper left entry is
zero, and upper left $3\times 3$ block is rank at most one. The
equations defining $\ol X_{2143}$ are the determinants
$$
\Big\< z_{11}, \quad
\det
\left|\begin{array}{ccc}z_{11}&z_{12}&z_{13}\\
z_{21}&z_{22}&z_{23}\\
z_{31}&z_{32}&z_{33}\end{array}\right|
= -z_{13} z_{22} z_{31} + \ldots
\Big\>.
$$
Note that this is {\em not} a Gr\"obner basis with respect to
term orders that pick out the diagonal term $z_{11} z_{22} z_{33}$ of
the second generator, since $z_{11}$ divides that.
The term orders that interest us pick out
the \emph{anti\/}diagonal term $-z_{13} z_{22} z_{31}$.
When we Gr\"obner-degenerate the matrix Schubert variety to the scheme
defined by the initial ideal $\$, we
get a union of three coordinate subspaces
$$
L_{11,13}, L_{11,22}, \hbox{ and } L_{11,31}, \quad\hbox{with
ideals}\quad \, \, \hbox{ and }
\.
$$
The resulting equation in $T$-equivariant cohomology, or on
multidegrees, reads:
$$
\begin{array}{cccccccccc}
[\ol X_w]
&=& [L_{11,13}] &+& [L_{11,22}] &+& [L_{11,31}]
\\ &=& \weight(z_{11}z_{13})&+&\weight(z_{11}z_{22})&+&\weight(z_{11}z_{31})
\\ &=& x_1^2&+&x_1x_2&+&x_1x_3
\end{array}
$$
in $\ZZ[x_1,x_2,x_3,x_4] \cong H^*_T(M_4)$.
%
Pictorially, we can represent the subspaces $L_{11,13}$, $L_{11,22}$,
and $L_{11,31}$ by drawing the $4 \times 4$ grid and placing a \cross
in each box containing a generator for its ideal---that is, place a
\cross at $(i,j)$ if every matrix in the subspace has $(i,j)$ entry
zero:
\begin{rcgraph}
\hbox{\normalsize{$\\ =\ $}}\
\begin{array}{|c|c|c|c|}
\multicolumn{4}{c}{}
\\[-2.5ex]
\hline\!+\!& &\!+\!&
\\\hline &\ \, & &\ \,
\\\hline & & &
\\\hline & & &
\\\hline
\end{array}
\:
\hbox{\normalsize{$,\quad\\ =\ $}}\
\begin{array}{|c|c|c|c|}
\multicolumn{4}{c}{}
\\[-2.5ex]
\hline\!+\!& &\ \, &\ \,
\\\hline &\!+\!& &
\\\hline & & &
\\\hline & & &
\\\hline
\end{array}
\:
\hbox{\normalsize{$,\quad\\ =\ $}}\
\begin{array}{|c|c|c|c|}
\multicolumn{4}{c}{}
\\[-2.5ex]
\hline\!+\!&\ \, &\ \, &\ \,
\\\hline & & &
\\\hline\!+\!& & &
\\\hline & & &
\\\hline
\end{array}
\end{rcgraph}
In the figures below, we draw the zero entries \cross by crossing
pipes, and the nonzero entries by elbow joints (imagine that the
lower-right triangle is filled with elbows).
$$
\qquad\qquad\
\begin{array}{ccccc}
&\perm1{}&\perm2{}&\perm3{}&\perm4{}\\
\petit2& \+ & \jr & \+ & \je \\
\petit1& \jr & \jr & \je &\\
\petit4& \jr & \je & &\\
\petit3& \je & & &\\
\end{array}
\qquad\qquad\hbox{}\qquad\
\begin{array}{ccccc}
&\perm1{}&\perm2{}&\perm3{}&\perm4{}\\
\petit2& \+ & \jr & \jr & \je \\
\petit1& \jr & \+ & \je &\\
\petit4& \jr & \je & &\\
\petit3& \je & & &\\
\end{array}
\qquad\qquad\hbox{}\qquad\
\begin{array}{ccccc}
&\perm1{}&\perm2{}&\perm3{}&\perm4{}\\
\petit2& \+ & \jr & \jr & \je \\
\petit1& \jr & \jr & \je &\\
\petit4& \+ & \je & &\\
\petit3& \je & & &\\
\end{array}
$$
These are the three ``rc-graphs'', or ``planar histories'', for the
permutation $2143$, and we recover the combinatorial formula from
\cite{FKyangBax,BB} for the Schubert polynomial $[\ol X_w]$ in the case
$w=2143$.
\end{example}
\subsubsection{Algebra}%Gr\"obner bases}
The Gr\"obner bases that arise during these computations have many
other applications. In general, the importance of Gr\"obner bases in
recent work on determinantal ideals and their relatives, such as powers
and symbolic powers, cannot be overstated. They are used in treatments
of questions about Cohen--Macaulayness, rational singularities,
multiplicity, dimension, $a$-invariants, and divisor class groups; see
\cite{CGG, StuGBdetRings, HerzogTrung, ConcaA-invariant, ConcaLadDet,
MotSoh1, MotSoh2, ConcaHerLadRatSing, BruConKRSandPowers,
GoncMilMixLadDet} for a sample. Since determinantal ideals as well as
their Gr\"obner bases also arise in the study of (partial) flag varieties
and their Schubert varieties (see Section~\ref{sub:other} and
\cite{BilLak, GonLakLadDetSchub, GonLakSchubToricLadDet,
GoncMilMixLadDet}, for instance), it is surprising to us that Gr\"obner
bases for the determinantal ideals defining matrix Schubert varieties
$\ol X_w$ do not seem to be in the literature, even though the ideals
themselves appeared in \cite{FulDegLoc}.
Our main results along these commutative algebraic lines are as follows.
Consider the $\ZZ^n$-grading on the polynomial ring $\kk[\zz]$ in $n^2$
variables $(z_{ij})_{i,j=1}^n$ over a field~$\kk$ in which $z_{ij}$ has
weight~$x_i$. The ideal $I_w$ of $\ol X_w$ is generated by determinants
and therefore homogeneous for this grading.
% In particular, $I_w$ is $\ZZ$-graded and hence has a well-defined degree.
% The Hilbert series and degree formulae in (\ref{hilb}) and~(\ref{deg})
% are really specializations of $\ZZ^n$-graded formulae. In particular,
% the numerator of the $\ZZ^n$-graded Hilbert series of $\ol X_w$ equals
% the Grothendieck polynomial $\GG_w(x_1,\ldots,x_n)$,
% \cite{LSgrothVarDrap},
% Of course, the determinants generating $I_w$ have integer coefficients,
% and therefore define a determinantal ideal in $\KK[\zz]$ over any
% field~$\KK$.
We prove:
\begin{textlist}
\item \label{deg}
The multidegree of the ideal $I_w$ is $\SS_w(x_1,\ldots,x_n)$,
the Schubert polynomial of Lascoux and Sch\"utzenberger.
% \cite{LSpolySchub}.
\item \label{gb}
The minors in $I_w$ form a Gr\"obner basis for any antidiagonal term
order. This result is new even for the determinantal ideals we found in
the literature, as they generally use diagonal term orders.
\item \label{hilb}
The $\ZZ^n$-graded Hilbert series of $\kk[\zz]/I_w$ is
$\GG_w(x_1,\ldots,x_n)/\prod_{i=1}^n(1-x_i)^n$, where the numerator is
the Grothendieck polynomial of Lascoux and Sch\"utzenberger.
\item \label{cm}
The initial complex $\LL_w$ is a shellable ball or sphere, and hence
Cohen--Macaulay.
\end{textlist}\setcounter{separated}{\value{enumi}}
The first statement is Theorem~\ref{thm:oracle}, and while quite close to
results that are well-known, our proof introduces some new techniques.
The next two statements appear in Theorem~\ref{thm:gb}, and the last is
Corollary~\ref{cor:shell} along with Corollary~\ref{cor:ball}. Formulae
for the $\ZZ$-graded Hilbert series and degree follow respectively
from~(\ref{deg}) by setting $(x_1,\ldots,x_n) = (1,\ldots,1)$, and
from~(\ref{hilb}) by setting $(x_1,\ldots,x_n) = (t,\ldots,t)$.
% All of these algebraic results hold with $\CC$ replaced by any field,
% having arbitrary characteristic.
It follows from (\ref{cm}) that $\ol X_w$ is Cohen--Macaulay, and an easy
consequence is a new proof of the theorem of Ramanathan
\cite{ramanathanCM} to the effect that Schubert varieties $X_w \subseteq
\FL$ are Cohen--Macaulay (Corollary~\ref{cor:cm}).
%
% {\bf This is the place to say we prove rationality of singularities, if
% we do.}
%
Conversely, we show that the product of $\ol X_w$ with a certain affine
space embeds as an open subvariety of a Schubert variety in $\FLN$ for
some $N \geq n$ (Corollary~\ref{cor:bigcell}), so the Cohen--Macaulayness
of $\ol X_w$ follows from that of Schubert varieties. More generally,
any local statement about the class of all Schubert varieties in flag
manifolds that remains valid after taking products with affine spaces is
equivalent to the corresponding statement for matrix Schubert varieties
(Theorem~\ref{thm:equiv}). Since rationality of singularities and
normality are among such local statements, $\ol X_w$ possesses these
properties because Schubert varieties do
\cite{ramanathanCM,RamRamNormSchub}.
Fulton used a similar (but different) argument in \cite{FulDegLoc} to
derive the Cohen--Macaulayness of $\ol X_w$, and could just as easily
have concluded the normality and rationality of singularities. We compare
his methods to ours and others in the latter parts of
Section~\ref{sec:app}.
% in Sections~\ref{sub:minimal}--\ref{sub:other}.
In particular, we observe in Section~\ref{sub:minimal} that Fulton's
notion of essential set for a permutation and the resulting
characterization of vexillary permutations identifies the popular class
of one-sided ladder determinantal varieties (see the references above) as
the class of vexillary matrix Schubert varieties
(Proposition~\ref{prop:vex}). Thus our results contain as special cases
many known and some unknown statements about one-sided ladder
determinantal ideals. For instance, the multidegree formula~(\ref{deg})
becomes completely explicit, taking the form of a determinantal
expression (Theorem~\ref{thm:vexdeg}). As we mentioned in~(\ref{gb}),
even the Gr\"obner basis theorem is new for ladder determinantal ideals,
because the term order differs from those appearing in the literature
(Section~\ref{sub:other}).
To prove~(\ref{deg})--(\ref{cm}), we introduce a collection of
geometric and combinatorial tools to the study of determinantal ideals
that we hope will be useful to algebraists in future investigations. At
present, ``almost all of the approaches one can choose for the
investigation of determinantal rings use standard bitableaux and the
straightening law'' \cite[p.~3]{BruConKRSandDet}, and are thus
intimately tied to the Knuth--Robinson--Schensted correspondence. However,
our Gr\"obner basis statements above do not seem to yield to KRS
techniques (Example~\ref{ex:13254} and the discussion in
Section~\ref{sub:other} explain one reason why). We find it necessary
instead to rely on our algebraic reformulation of equivariant cohomology
and $K$-theory of flag manifolds and related spaces, as well as on
combinatorial methods involving rc-graphs, antidiagonals,
% Alexander duality,
Stanley--Reisner theory, and a new class of shellable simplicial
complexes defined using subwords of a fixed word in a Coxeter group.
For example, we calculate the multidegree of $\ol X_w$ using an inductive
technique that, while based on calculations using localization of
equivariant cohomology to torus-fixed points, is nonetheless entirely
commutative algebraic, requiring no localization theorems. (An
alternative calculation for the degree of $\ol X_w$, in
Appendix~\ref{app:basics}, proceeds by comparing the ordinary cohomology
of $\FL$ to the equivariant cohomology of $\mn$ for the torus action of
left multiplication by diagonal invertible matrices.) The Hilbert series
calculation for $\LL_w$ in Sections~\ref{sub:lift} and~\ref{sub:coarsen},
which is fundamental to our proof of (\ref{gb}), is really a computation
in equivariant $K$-theory, as demonstrated by Corollary~\ref{cor:groth}:
$\GG_w$ represents the $K$-theory class of $X_w$ on $\FL$
\cite{LSgrothVarDrap,LascouxGrothFest}. In fact, though, the Hilbert
series calculation proceeds via the combinatorics of the antidiagonals
that generate the initial ideal
(Sections~\ref{sub:lift},~\ref{sub:coarsen}, and~\ref{sub:Lw}). The
Stanley--Reisner theory in Section~\ref{sub:alex} then enables the
multidegree calculation for~$\LL_w$.
\subsubsection{Combinatorics}%al constructions}
The arguments of Section~\ref{sec:rc} involved in the proof of
Cohen--Macaulayness of the initial complex~(\ref{cm}) actually benefit
the combinatorics as much as they do the algebra and geometry. Our main
results here include the following.
\begin{textlist}\setcounter{enumi}{\value{separated}}
\item \label{rc}
The facets~$\LL_w$ are in natural bijection with the rc-graphs for~$w$ of
Fomin and Kirillov \cite{FKyangBax}.
\item \label{bjs}
Our positive formula for Schubert polynomials agrees with---and provides
a new geometric proof/explanation of---the combinatorial formula of
Billey, Jockusch, and Stanley \cite{BJS}.
\item \label{mitosis}
There is a simple combinatorial rule, called \bem{mitosis}, to list all
of the rc-graphs for a given permutation positively (i.e.\ without
cancellation) by induction on the weak Bruhat order.
\item \label{subword}
The simplicial complex $\LL_w$ is a \bem{subword complex} in the Coxeter
group $S_n$, and is therefore a vertex-decomposable ball or sphere.
\end{textlist}\setcounter{separated}{\value{enumi}}
The rc-graphs for a permutation in $S_n$ are subsets of $[n] \times [n]$,
and can be thought of as a generalization to flag manifolds of
semistandard Young tableaux for Grassmannians, there being a natural
bijection between tableaux and rc-graphs for Grassmannian permutations
(as can be found in \cite{KoganThesis}, for instance).
Our argument for Theorem~\ref{thm:rc}, which contains the statement
of~(\ref{rc}), uses the proof of~(\ref{gb}) along with combinatorial
results of Bergeron and Billey \cite{BB}. We then prove~(\ref{bjs}) as a
consequence of~(\ref{rc}) and~(\ref{deg}). (Another proof
of~(\ref{bjs}), based on symplectic geometry, appears in
\cite{KoganThesis}.)
The combinatorics that goes into our proof of the Gr\"obner basis
theorem~(\ref{gb}) in Section~\ref{sec:gb} translates via~(\ref{rc})
into~(\ref{mitosis}), which appears as Theorem~\ref{thm:mitosis}.
Mitosis serves as a geometrically motivated substitute for Kohnert's
conjecture \cite{Kohnert, NoteSchubPoly, Winkel}. To demonstrate its
simplicity, we outline in Section~\ref{sub:combin} a self-contained
combinatorial proof using rc-graphs and the formula of \cite{BJS}, in
addition to our earlier proof via antidiagonals and Gr\"obner bases.
% I am in favor of this extra computation because I think the
% combinatorialists need convincing that a simple alternative to
% Kohnert's rule exists, and because they'd feel a lot better about the
% proof knowing they only need to work with rc-graphs and not these new
% antidiagonals.
The origin of~(\ref{subword}) was our desire, at the outset, to find some
existing family of Cohen--Macaulay complexes containing~$\LL_w$. The
seeming absence of such a family in the literature motivated us instead
to define here the new class of subword simplicial complexes, containing
$\LL_w$ as a special case. Given a Coxeter group $\Pi$, there is a
subword complex associated to each pair consisting of a word in $\Pi$ and
an element of~$\Pi$. The structures governing subword complexes relax
the axioms for greedoids in a manner that still preserves the fundamental
properties of the dual greedoid complex. In particular,
Theorem~\ref{thm:shell} says that subword complexes are
vertex-decomposable, and hence shellably Cohen--Macaulay. In fact, more
is true: Theorem~\ref{thm:ball} says that subword complexes are
homeomorphic to balls or spheres, and identifies their boundary faces (if
there are any).
Sections~\ref{sub:schub}, \ref{sub:rc}, and~\ref{sub:links} contain our
main applications of the results in Section~\ref{sec:gb} to
% geometry and formulae for the coefficients of
facts concerning Schubert and Grothendieck polynomials that are mostly
known, but find here either generalizations or new proofs (or both). In
addition to the motivational considerations already mentioned at the
beginning of this Introduction, we find particularly intriguing the
interaction with the Eagon-Reiner theorem \cite{ER} to $\GG_w$ in
Section~\ref{sub:links}: in concert with the Gr\"obner basis theorem, the
Alexander inversion formula (Proposition~\ref{prop:inversion}), and the
Cohen--Macaulayness of $\LL_w$, it implies that the coefficients on each
homogeneous piece of $\GG_w(\1-\xx)$ all have the same sign
(Remark~\ref{rk:ER}). We derive a combinatorial formula for these
coefficients (Corollary~\ref{cor:links}) in terms of nonreduced
expressions and compatible sequences, which we call pipe dreams
(Definition~\ref{defn:pipe}). This formula, which agrees with that of
Fomin and Kirillov \cite{FKgrothYangBax}, is a special case of a
computation for subword complexes (Theorem~\ref{thm:links}) achieved by
applying Theorems~\ref{thm:shell} and~\ref{thm:ball} along with
Hochster's theorem from Stanley-Reisner theory concerning Betti numbers
for squarefree monomial ideals.
\subsubsection{Geometry}%ic applications}
Sections~\ref{sub:schub} and~\ref{sub:groth} relate our algebraic and
combinatorial methods in Section~\ref{sec:gb} to the geometry of Schubert
polynomials and Grothendieck polynomials. In particular, this is where
we
\begin{textlist}\setcounter{enumi}{\value{separated}}
\item \label{schub}
explain the positivity of Schubert coefficients by identifying the
intrinsic geometric objects counted by their coefficients; and
\item \label{groth}
recover the role of Grothendieck polynomials as the $K$-theory classes of
structure sheaves of Schubert varieties on the flag manifold
\cite{LSgrothVarDrap}.
\end{textlist}\setcounter{separated}{\value{enumi}}
We have already said a good deal about~(\ref{schub}). One interesting
point about our derivation of~(\ref{groth}) is that it requires no
assumptions about the rational singularities of Schubert varieties: the
multidegree proof of the Hilbert series calculation~(\ref{hilb}), which
is based on cohomological considerations that ignore phenomena at complex
codimension~$1$ or more, automatically produces the $K$-classes as
numerators of Hilbert series.
Section \ref{sub:loci} contains an explanation of how to
\begin{textlist}\setcounter{enumi}{\value{separated}}
\item \label{loci}
derive the connection between matrix Schubert varieties and Fulton's
theory of degeneracy loci for maps between flagged vector bundles.
\end{textlist}
This is due to the double appearance of $BB \times BB_+$ as the base
space for $(B\times B_+)$-equivariant cohomology, and as the classifying
space for pairs of flagged vector bundles. Borel's mixing construction
from the equivariant cohomology context, when applied to $\ol X_w$, gives
the universal degeneracy locus inside the universal $\Hom$-bundle from
the classifying space context. Since completing this work, we learned of
the nice paper \cite{FRthomPoly} and refer the reader to it for greater
detail about the topological considerations and applications (as
Section~\ref{sub:loci} is rather a departure from the commutative algebra
framework underlying the rest of our exposition).
Everything we have to say about Lie groups concerns type~$A$. While the
definition of Schubert {\em class} is clear for arbitrary groups $G$, the
definition of Schubert {\em polynomial} is less clear for groups other
than $\gln$ \cite{FKBn}, and we do not have anything to say here about
their geometry. In particular, some flavors do not have positive
coefficients, and so cannot arise from Gr\"obner degenerations to
(automatically) effective classes.
% {{\bf check out Lex Renner and Mohan Putcha for reductive monoids}}
\paragraph*{Prerequisites.}
We have tried to make the material below as accessible as possible to
combinatorialists, geometers, and commutative algebraists alike.
Therefore, when there was a choice of whether to include or omit the
reason for a basic fact, we usually included~it. In particular, we have
assumed no specific knowledge of the geometry or combinatorics of flag
manifolds, or of Schubert varieties, Schubert polynomials, and
Grothendieck polynomials. The remaining parts of this Introduction aim
to fill in this background, as seen from our point of view using
multidegrees.
% for the reader's convenience we provide a number
Other background material can be found in the appendices on: equivariant
$K$-theory, multigradings, and Hilbert series; the new algebraic
machinery of multidegrees; the equivariant cohomology on which the
intuition for the latter is based, in the situations to which we apply
it; some background on the topology and algebraic geometry of flag
manifolds, including a geometric argument for~(\ref{deg}) and~(\ref{gb})
above; and finally, term orders, Gr\"obner bases, and initial ideals.
Many of our examples interpret the same underlying data in varying
contexts, to highlight common themes. In particular this is true of
Examples~\ref{ex:intro}, \ref{ex:mutate}, \ref{ex:prom}, \ref{ex:ddem},
\ref{ex:ii}, \ref{ex:revert}, \ref{ex:ess}, \ref{ex:pipe},
\ref{ex:mitosis}, \ref{ex:mitosis'}, \ref{ex:remove}, \ref{ex:rc},
and~\ref{ex:apoptosis}.
\paragraph*{Acknowledgements.}
The authors are grateful to Bernd Sturmfels, who took part in the genesis
of this project, and to Misha Kogan, as well as to Sara Billey, Anders
Buch, Cristian Lenart, Vic Reiner, Rich\'ard Rim\'anyi, Anne Schilling,
Frank Sottile, and Richard Stanley for inspiring conversations and
references. Nantel Bergeron kindly provided \LaTeX\ macros for
drawing pipe dreams.
%AK was
%funded by the Clay Mathematics Institute for part of this project, the
%Alfred P. Sloan Foundation, and the NSF.
%EM was funded by the Alfred P. Sloan Foundation during the earlier stages of
%this project, and by the NSF thereafter.
\subsection{Multidegrees and squarefree monomial ideals}\label{sub:alex}%
We review here some basic facts of Stanley--Reisner theory concerning
squarefree monomial ideals. These facts then help us define and justify
the notion of multidegree for the ideals and subschemes appearing in the
main body of this monograph. The general introduction to multidegrees
can be found in Appendix~\ref{app:degrees}.
Since Lemma~\ref{lemma:inversion} will be applied in
Section~\ref{sub:links} to squarefree monomial ideals in any number of
variables, we begin by working with a $\ZZ^m$-graded polynomial ring
$\kk[\zz] = \kk[z_1, \ldots, z_m]$, although the reader may assume $m =
n^2$ for the applications to the antidiagonal ideals~$J_w$ (introduced in
Section~\ref{sub:gb}). The field~$\kk$ can have arbitrary
characteristic, and can be assumed algebraically closed for convenience,
although this hypothesis is unnecessary if one considers things like
vector subspaces and orbits of group actions in their natural
scheme-theoretic sense.
For notation, we write $\ZZ^m$-graded Hilbert series in the variables
$\zz = z_1, \ldots, z_m$. The Hilbert series of any $\ZZ^m$-graded
$\kk[\zz]$-module $\Gamma$ satisfies
\begin{eqnarray*}
H(\Gamma;\zz) &=& \frac{\KK(\Gamma;\zz)}{\prod_{i=1}^m(1-z_i)}
\end{eqnarray*}
for the \bem{Hilbert numerator} Laurent polynomial $\KK(\Gamma;\zz)$, as
in Proposition~\ref{prop:numerator} of the first appendix. For a
coordinate subspace $L \subseteq \kk^m$, let $D_L \subseteq \{1, \ldots,
m\}$ be the subset such that the ideal $\$ consists
of those functions vanishing on~$L$. Finally, set $\zz^D = \prod_{i \in
D} z_i$ for $D \subseteq \{1, \ldots, m\}$.
\begin{lemma} \label{lemma:inversion}
Suppose that the squarefree monomial ideal $J$ has zero set $\LL$, and
that $\kk[\zz]/J$ has $\ZZ^m$-graded Hilbert numerator $\KK(\zz)$. If
$\cC(\zz)$ is the sum of lowest weight terms in the polynomial
$\KK(\1-\zz)$ obtained from $\KK(\zz)$ by substituting $1-z_i$
\hbox{for~$z_i$, then}
\begin{eqnarray*}
\cC(\zz) &=& \sum \zz^{D_L},
\end{eqnarray*}
where the sum is over all subspaces $L \subseteq \LL$ of maximal
dimension.
\end{lemma}
By ``the sum of lowest weight terms'' here, we mean the homogeneous
component (in the usual sense) of minimal total degree.
\begin{proof}
By definition, $J = \bigcap_{L \in \LL} \$. The
Hilbert series of $\kk[\zz]/J$ is therefore the sum of all monomials
outside every one of the ideals $\$ for coordinate
subspaces $L \subseteq \LL$. This sum is over the monomials $\zz^\bb$
for $\bb \in \ZZ^m$ having support exactly $\{1,\ldots,m\} \minus D_L$
for some $L \subseteq \LL$:
\begin{eqnarray} \label{eq:J}
H(\kk[\zz]/J;\zz) &=& \sum_{L \in \LL} \prod_{i \not\in D_L}{z_i \over
1-z_i}\ \:=\ \:\sum_{L \in \LL} {\prod_{i \not\in D_L}(z_i) \prod_{i
\in D_L} (1-z_i) \over \prod_{i=1}^m(1-z_i)}.
\end{eqnarray}
After substituting $1-z_i$ for $z_i$ in the last expression above, the
lowest weight terms in the numerator correspond to the subspaces $L
\subseteq \LL$ of highest dimension.
\end{proof}
Many of our applications concern gradings that aren't as fine as the
$\ZZ^m$-grading. Specifically, let $m = n^2$, so that $\zz =
(z_{ij})_{i,j=1}^n$ and $\kk[\zz]$ is the coordinate ring of the $n
\times n$ matrices $\mn$ over~$\kk$. Consider the following two gradings
on $\kk[\zz]$, in addition to the $\ZZ^{n^2}$-grading:
\begin{itemize}
\item
the $\ZZ^n$-grading in which the \bem{weight} of the variable $z_{ij}$ is
defined by \hbox{$\weight(z_{ij}) = x_i$}, where $\xx = x_1,\ldots,x_n$
is the standard basis of $\ZZ^n$; and
\item
the $\ZZ^{2n}$-grading in which $\weight(z_{ij}) = x_i-y_j$, where $\yy =
y_1,\ldots,y_n$ is the standard basis for another copy of $\ZZ^n$.
\end{itemize}
In the next subsection, where we justify geometrically why we consider
these gradings, we shall also have occasion to consider the
$\ZZ^{2n}$-graded polynomial ring $\kk[\zz][u]$ in $n^2 + 1$ variables,
where the extra variable $u$ has weight $x_i - x_{i+1}$.
% Also, for a brief stint in Lemma~\ref{lemma:IN}, the usual
% $\ZZ$-grading (in which every variable has weight~$t$, where $t$ is the
% standard basis for~$\ZZ$) will appear.
Consider one of the polynomial rings $R = \kk[\zz]$ or $\kk[\zz][u]$,
with one of the multigradings above. These are all positively graded (as
in Appendix~\ref{app:K-theory}), so the graded pieces of $R$ have finite
dimension. We write multigraded Hilbert series of multigraded quotients
of $R$ using the $\xx$ and $\yy$ variables, so that (for instance), the
Hilbert series of a $\ZZ^{2n}$-graded quotient $R/I$ of $R = \kk[\zz][u]$
always has the form
\begin{eqnarray*}
H(R/I;\xx,\yy) &=& \frac{\KK(R/I;\xx,\yy)}{(1-{x_i}/{x_{i+1}})
\prod_{i,j=1}^n(1-{x_i}/{y_j})}.
\end{eqnarray*}
The Hilbert numerator $\KK(R/I;\xx,\yy)$ here is a Laurent polynomial,
while the factors in the denominator correspond to the variables in $R$,
which is $\kk[\zz][u]$ in this case. Note that the weights of the
variables show up in the denominator in exponential form: the
\bem{exponential weight} $x_i/y_j$ corresponds to the variable $z_{ij}$
of weight $x_i - y_j$.
Transition between the exponential and ordinary forms of the weights can
be accomplished by analogy with the usual exponential function: expanding
$1-e^{-x}$ as a power series in $x$ and taking lowest degree terms yields
back~$x$ again. (This analogy is explained in Remark~\ref{rk:quirk} as
an algebraic reflection of the Poincar\'e homomorphism and the
Riemann--Roch isomorphism of cohomology with the associated graded ring
of $K$-theory.) In the case of an arbitrary Laurent polynomial
$\KK(\xx,\yy)$, the rational function $\KK(\1-\xx,\1-\yy)$, in which each
$x_i$ has been replaced by $1-x_i$ and each $y_j$ by $1-y_j$, can be
expanded as a well-defined power series in $\xx$ and $\yy$.
\begin{defn} \label{defn:sqfreedeg}
Let $R = \kk[\zz]$ or $\kk[\zz][u]$ as above. The $\ZZ^n$-graded or
$\ZZ^{2n}$-graded \bem{multidegree} of a respectively $\ZZ^n$-graded or
$\ZZ^{2n}$-graded quotient $R/I$ is the sum $[\spec(\kk[\zz]/I)]_{\ZZ^n}$
or $[\spec(R/I)]_{\ZZ^{2n}}$ of all lowest weight terms in the
substituted Laurent polynomial $\KK(\kk[\zz]/I;\1-\xx)$ or
$\KK(R/I;\1-\xx,\1-\yy)$.
% for the corresponding multigrading.
\end{defn}
% To be clear, both the bracket notation and the $\cC$ notation are being
% defined here---that is, the equalities are not consequences of some
% previous lemma.
The brackets are supposed to suggest some sort of cohomology class; see
Remark~\ref{rk:letters}. The subscript $\ZZ^{..n}$ might be dropped, if
the grading is either clear from context or irrelevant. It will
sometimes be useful to have the variables $\xx$ and possibly $\yy$
explicitly in the notation. When required, we will instead write
\begin{eqnarray*}
[\spec(\kk[\zz]/I)]_{\ZZ^n} = \cC(\kk[\zz]/I;\xx) &\hbox{ or }&
[\spec(R/I)]_{\ZZ^{2n}} = \cC(R/I;\xx,\yy).
\end{eqnarray*}
\begin{example} \label{ex:subspace}
The Hilbert numerator of $\kk = \kk[\zz]/\$
is the universal denominator: $\KK(\kk;\xx,\yy) =
\prod_{i,j=1}^n(1-{x_i}/{y_j})$. The multidegree of $\kk$ is the product
$\cC(\kk;\xx,\yy) = \prod_{i,j=1}^n (x_i-y_j)$. The $\ZZ^n$-graded
versions of these calculations, $\KK(\kk;\xx) = \prod_{i=1}^n(1-{x_i})^n$
and $\cC(\kk;\xx) = \prod_{i=1}^n x_i^n$, are obtained from the
$\ZZ^{2n}$ versions by setting all of the $\yy$ variables to $1$ and~$0$,
respectively.
If a subspace $L \subseteq \mn$ is the zero set of $\$, then its multidegree is
\begin{eqnarray} \label{eq:wt}
[L] &=& \prod_{(i,j) \in D_L} \weight(z_{ij}).
\end{eqnarray}
The absence of the $\ZZ^{..n}$ subscript is useful here, as~(\ref{eq:wt})
holds for any multigrading.%
\end{example}
The next two results generalize the observations in the previous example.
to the point where they become useful in later sections; the second is in
fact applied directly in the proof of Theorem~\ref{thm:gb}.
\begin{lemma} \label{lemma:coarsen}
If $I \subseteq \kk[\zz]$ is a $\ZZ^{2n}$-graded ideal, then $I$ is also
$\ZZ^n$-graded. The $\ZZ^n$-graded and $\ZZ^{2n}$-graded Hilbert series
and multidegrees of $\kk[\zz]/I$ satisfy
\begin{eqnarray*}
H(\kk[\zz]/I;\xx) = H(\kk[\zz]/I;\xx,\1) &\hbox{ and }&
\cC(\kk[\zz]/I;\xx) = \cC(\kk[\zz]/I;\xx,\0).
\end{eqnarray*}
\end{lemma}
\begin{proof}
The statement about Hilbert series follows by thinking of them as
generating functions for the number of monomials having given weight.
The multidegree statement is an easy consequence, as it can be checked on
a single Laurent monomial.%
\end{proof}
\begin{prop} \label{prop:degree}
If some squarefree monomial ideal has zero set $\LL$, then
\begin{eqnarray*}
[\LL] &=& \sum_\twoline{L \subseteq \LL}{\dim L = \dim\LL} [L]
\end{eqnarray*}
in any grading; i.e.\ sum expressions~(\ref{eq:wt}) over subspaces $L
\subseteq \LL$ of \hbox{maximal dimension}.
\end{prop}
\begin{proof}
Translate the Hilbert series~(\ref{eq:J}) into the $m = n^2$ notation,
and use the $\ZZ^n$- or $\ZZ^{2n}$-grading. Now substitute $\xx \mapsto
\1-\xx$ and $\yy \mapsto \1-\yy$ and calculate.%
\end{proof}
Proposition~\ref{prop:degree} is easier than the corresponding general
result, Proposition~\ref{prop:hilbdeg}, precisely because we have an
explicit formula for the Hilbert series in~(\ref{eq:J}). It justifies
the geometric content of multidegree for nonmonomial ideals by taking
initial ideals:
\begin{cor} \label{cor:grobdeg}
If the multigraded ideal $I \subseteq \kk[\zz]$ of a subscheme $X
\subseteq \mn$ has initial ideal $J$ for some term order, then
$\cC(\kk[\zz]/I;\xx,\yy) = \cC(\kk[\zz]/J;\xx,\yy)$. In particular, if
$J$ is a squarefree monomial ideal with zero set~$\LL$, then
\begin{eqnarray*}
[X] &=& \sum_\twoline{L \subseteq \LL}{\dim L = \dim\LL} [L].
\end{eqnarray*}
\end{cor}
\begin{proof}
Hilbert series, and thus multidegrees, are preserved by taking intial
ideals.%
\end{proof}
There is one final result we need concerning multidegrees.
Unfortunately, its proof requires the full machinery of
Appendix~\ref{app:degrees}, and thus fails to fit the flow of this
introductory section. Therefore, we reluctantly banish its proof to
% Appendix~\ref{app:degrees}
Corollary~\ref{cor:additive}. The result is used twice in the main text,
in the proofs of Theorem~\ref{thm:oracle} and Lemma~\ref{lemma:IN}. For
all of our other applications, Corollary~\ref{cor:grobdeg} suits us
better.
\begin{prop} \label{prop:additive'}
Let $R = \kk[\zz][u]$ or $\kk[\zz]$, and $I \subseteq R$ is a
$\ZZ^{2n}$-graded ideal.
% Suppose $I = I' \cap I_1\cap\cdots\cap I_r$, where $\dim(R/I') <
% \dim(R/I)$ and no two $I_\ell$ share an associated prime.
If $I_1,\ldots,I_r$ are the top-dimensional primary components of~$I$,
then
\begin{eqnarray*}
\cC(R/I;\xx,\yy) &=& \sum_{\ell=1}^r \cC(R/I_\ell;\xx,\yy).
\end{eqnarray*}
% the subscheme $X$ defined by $I$ can be expressed as a union $X = X_1
% \cup \ldots \cup X_\ell$ of
\end{prop}
\begin{remark} \label{rk:letters}
The letters $\cC$ and $\KK$ throughout this text stand for `cohomology'
and \hbox{`$K$-theory'}. When $\kk = \CC$ is the complex numbers, the
(Laurent) polynomials they denote are honest torus-equivariant cohomology
and $K$-classes on affine space. Since affine space is equivariantly
contractible, the equivariant cohomology and $K$-theory rings are
naturally polynomial and Laurent polynomial rings over $\ZZ$,
respectively. Had we defined the $\ZZ$-graded multidegree (which we
don't need here, surprisingly, but appears in the Appendix), we would see
that the $\ZZ$-graded multidegree of a $\ZZ$-graded ideal equals the
usual degree of its corresponding projective scheme times~$t^{\rm
codim}$. See Appendix~\ref{app:degrees} for details. In particular, see
Remark~\ref{rk:quirk} for the history behind the odd practice of using
$x_i$ as both an exponential and ordinary variable.%
\end{remark}
\subsection{Matrix Schubert varieties}\label{sub:matschub}%%%%%%%%%%%%%%%
Let $\mn$ be the $n \times n$ matrices over $\kk$, with coordinate ring
$\kk[\zz]$ in indeterminates $\{z_{ij}\}_{i,j = 1}^n$. Throughout the
paper, $q$ and $p$ will be integers with $1 \leq q,p \leq n$, and $Z$
will stand for an $n \times n$ matrix. Most often, $Z$ will be the
\bem{generic matrix} of variables $(z_{ij})$, although occasionally $Z$
will be an element of $\mn$. Denote by $Z\sub qp$ the northwest $q
\times p$ submatrix of $Z$. Given a permutation $w \in S_n$, for
instance, we find that
\begin{eqnarray*}
\rank(w^T\sub qp) &=& \#\{(i,j) \leq (q,p) \mid w(i) = j\}
\end{eqnarray*}
is the number of $1$'s in the submatrix $w^T\sub qp$ (recall that the
$1$'s lie at $w^T_{i,w(i)}$).
The following definition was made by Fulton in \cite{FulDegLoc}.
\begin{defn} \label{defn:matrixSchub}
%\begin{defnlabeled}{{\cite[Section~3]{FulDegLoc}}}\end{defnlabeled}
Let $w \in S_n$.
The \bem{matrix Schubert variety} $\ol X_w \subseteq \mn$ consists of the
matrices $Z \in \mn$ such that $\rank(Z\sub qp) \leq \rank(w^T\sub qp)$
for all $q,p$.
\end{defn}
\begin{example} \label{ex:w0}
The smallest matrix Schubert variety is $\ol X_{w_0}$, where $w_0$ is the
\bem{long permutation} \hbox{$n\,\cdots\,2\,1$} reversing the order of
$1,\ldots,n$. The variety $\ol X_{w_0}$ is just the linear subspace of
lower-right-triangular matrices whose ideal is $\$.
\end{example}
\begin{example} \label{ex:s3}
Five of the six $3\times 3$ matrix Schubert varieties are linear
subspaces:
\begin{eqnarray*}
\ol X_{123} &=& M_3
\\
\ol X_{213} &=& \{Z \in M_3\,|\, z_{11} = 0\}
\\
\ol X_{231} &=& \{Z \in M_3\,|\, z_{11} = z_{12} = 0\}
\\
\ol X_{312} &=& \{Z \in M_3\,|\, z_{11} = z_{21} = 0\}
\\
\ol X_{321} &=& \{Z \in M_3\,|\, z_{11} = z_{21} = z_{12} = 0\}
\end{eqnarray*}
The last one, $\ol X_{132}$, is the set of matrices
\begin{eqnarray*}
\ol X_{132} &=& \{Z \in M_3\,|\, z_{11}z_{22} - z_{12}z_{21} = 0\}\,
\end{eqnarray*}
whose upper left $2\times 2$ block is singular.
\end{example}
Matrix Schubert varieties are clearly stable under rescaling any row or
column. Moreover, since we only impose rank conditions on submatrices
that are as far north and west as possible, any operation that adds a
multiple of some row to a row below it (``sweeping downward''), or that
adds a multiple of some column to another column to its right (``sweeping
to the right'') preserves every matrix Schubert variety.
These observations have useful group-theoretic restatements. Let $B$
denote the group of invertible {\em lower} triangular matrices and $B_+$
the invertible upper triangular matrices, which intersect in the
invertible diagonal matrices $T = B\cap B_+$. The previous paragraph
says exactly that each matrix Schubert variety $\ol X_w$ is preserved by
the action%
%
\footnote{This is a left group action, in the sense that
$(b,b_+) \cdot ((b',b'_+) \cdot Z)$ equals
$((b,b_+)\cdot(b',b'_+))\cdot Z$ instead of
$((b',b'_+)\cdot(b,b_+))\cdot Z$, even though---in fact
because---the $b_+$ acts via its inverse \hbox{on the {\em right}}.}
%
of $B\times B_+$ on $\mn$ in which $(b,b_+) \cdot Z = bZb_+^{-1}$.
Proposition~\ref{prop:BwB} will actually say much more.
The next three lemmas, which we require for Proposition~\ref{prop:BwB},
are basically standard results on Bruhat order for~$S_n$, enhanced
slightly for $\mn$ instead of~$\gln$; we provide proofs
% for completeness.
to remain completely self-contained. Recall that a \bem{partial
permutation matrix} $Z \in \mn$ has at most one~$1$ in each row and
column, and all remaining entries equal to zero.
\begin{lemma} \label{lemma:orbits}
In each $B \times B_+$ orbit on $\mn$ lies a unique partial permutation
matrix.
\end{lemma}
\begin{proof}
By doing row and column operations that sweep down and to the right, we
can get from an arbitrary matrix~$Z'$ to a partial permutation
matrix~$Z$.
% Thus there are only finitely many $B \times B_+$ orbits on~$\mn$,
% because there are only finitely many partial permutation matrices.
Such sweeping preserves the ranks of northwest $q \times p$ submatrices,
and~$Z$ can be reconstructed uniquely by knowing only $\rank(Z'\sub qp)$
for $1 \leq q,p \leq n$.
\end{proof}
By definition, the \bem{length} of a partial permutation matrix~$Z$ is
the number of zeros in $Z$ that lie neither due south nor due east of
a~$1$. In other words, for every $1$ in~$Z$, cross out all the boxes
beneath it in the same column as well as to its right in the same row,
and count the number of uncrossed-out boxes to get the length of~$Z$.
When $Z = w^T$ is a permutation matrix, define $\length(w)$ to be the
length of~$w^T \in \mn$. Write $Z \subseteq Z'$ for partial permutation
matrices $Z$ and~$Z'$ if the $1$'s in $Z$ are a subset of the $1$'s
in~$Z'$. Also, to simplify notation, write $|Z\sub qp|$ instead of
$\rank(Z\sub qp)$. Finally, let $t_{i,i'} \in S_n$ be the transposition
switching $i$ and~$i'$.
\begin{lemma} \label{lemma:bruhat}
Fix a permutation $w \in S_n$ and a partial permutation matrix~$Z$. If
$Z \in \ol X_w$ and $\length(Z) \leq \length(w)$, then either $Z
\subseteq w^T$, or there is a transposition $t_{i,i'}$ such that $v =
wt_{i,i'}$ satisfies: $Z \in \ol X_v$ and\/ $\length(v) > \length(w)$.
\end{lemma}
\begin{proof}
The proof is by downward induction on the number of $1$'s shared by $w^T$
and~$Z$, the case where this number is $n$ (so $Z = w^T$) being trivial.
Let $i$ be the first row in which $w^T$ has a $1$ where $Z$ does not, and
let $j = w(i)$ be the column index of this~$1$. Comparing rank
conditions along row~$i$, this means that either (a)~row~$i$ of $Z$ is
blank, or (b)~the~$1$ in row~$i$ of $Z$ lies in some column~$j'$ to the
right of column~$j$.
In case~(b), choose $i'$ satisfying $w(i') = j'$, so that $w^T$ has a~$1$
at $(i',j')$. Noting that $i < i'$ by minimality of~$i$, switch rows $i$
and $i'$ of $w^T$ to get~$v^T$, so that $v = wt_{i,i'}$. We leave to the
reader the standard check of the following statement.
\begin{claim} \label{claim:length}
% Suppose $(i,j) \leq (i',j')$, with $w(i) = j$ and $w(i') = j'$. Then
% the length of $v = wt_{i,i'}$ is $\length(w) + 1$ plus twice the number
% of~$1$'s strictly inside the rectangle enclosed by $(i,j)$ and
% $(i',j')$.\hfill$\Box$
Suppose $Z$ is a partial permutation matrix with $1$'s at $(i,j)$ and
$(i',j')$, where $(i,j) \leq (i',j')$. Switching rows $i$ and $i'$
of~$Z$ creates a partial permutation matrix $Z'$ satisfying\/
$\length(Z') = \length(Z) + 1 + \mbox{}$twice the number of~$1$'s strictly
inside the rectangle enclosed by $(i,j)$ and $(i',j')$.\hfill$\Box$
\end{claim}
In case~(a), one of two things can happen: (a$'$) there is some spot
$(i',j')$ strictly southeast of $(i,j)$ such that $w(i') = j'$; or
(a$''$) not. In case (a$'$), choose $i'$ minimal and switch rows $i$ and
$i'$ of $w^T$---that is, take $v = wt_{i,i'}$. Compare ranks to show $Z
\in \ol X_v$:
\begin{itemize}
\item
In any spot $(q,p)$ not satisfying $(i,j) \leq (q,p) \leq (i'-1,j'-1)$,
we have $|Z\sub qp| \leq |w^T\sub qp| \leq |v^T\sub qp|$, the first
inequality by definition and the second by construction.
\item
Setting $q = i$ and assuming $j \leq p \leq j'-1$ yields $|Z\sub ip| =
|Z\sub {i-1}p| \leq |w^T\sub {i-1}p| = |w^T\sub ip| - 1 = |v^T\sub ip|$.
%
\comment{the first equality because the $1$ in row $i$ of $Z$
occurs east of column~$p$; the inequality by definition; the
penultimate equality because the $1$ in row~$i$ of $w^T$ lies
west of (or in) column~$j$; and the last equality by
construction.}%
%
\item
If $i+\ell = q \leq i'-1$ for some $\ell \geq 0$, and $j \leq p \leq
j'-1$, then $|Z\sub qp| \leq |Z\sub ip| + \ell \leq |v^T\sub ip| + \ell =
|w^T\sub qp|$. The second inequality is the previous item, and the last
equality is because all the $1$'s in rows strictly between $i$ and $i'$
of $w^T$ lie to the left of column~$j$, by minimality of~$i'$.
\end{itemize}
In case (a$''$), we may replace $Z$ by the matrix $Z'$ obtained by adding
a~$1$ to the $(i,j)$ spot of~$Z$. The new matrix $Z'$ still lies in~$\ol
X_w$, by checking the number of $1$'s in $Z\sub qp$:
\begin{itemize}
\item
If $p < j$ or $q < i'$ then $Z\sub qp = Z'\sub qp$; that is, $Z$ and $Z'$
agree strictly north of row~$i$ and strictly west of column~$j$.
\item
Assuming $q \geq i'$ and setting $p = j$, we get $|Z\sub qj| = |Z\sub
q{j-1}| \leq |w^T\sub q{j-1}| = |w^T\sub qj| - 1$, and it follows that
$|Z'\sub qj| \leq |w^T\sub qj|$.
\item
If $q \geq i'$ and $p = j + \ell$ for some $\ell \geq 0$, then $|Z'\sub
qp| \leq |Z'\sub q{j}| + \ell \leq |w^T\sub q{j}| + \ell = |w^T\sub qp|$,
where the last equality is because all the $1$'s in columns east of~$j$
in $w^T$ lie north of row~$i$.
\end{itemize}
As $Z'$ shares more $1$'s with $w^T$ than $Z$,
% and $Z' \in \ol X_v \implies Z \in \ol X_v$,
induction completes the proof.%
\end{proof}
\begin{lemma} \label{lemma:Z}
Let $Z$ be a partial permutation matrix with orbit closure $\ol\OO_Z$ in
$\mn$. If\/ $\length(s_i Z) < \length(Z)$, then
$$
n^2 - \length(s_i Z)\ \:=\ \:\dim(\ol\OO_{s_i Z})\ \:=\ \:
\dim(\ol\OO_Z) + 1\ \:=\ \:n^2 - \length(Z) + 1,
$$
and $s_i(\ol \OO_{s_i Z}) = \ol\OO_{s_i Z}$.
\end{lemma}
\begin{proof}
Let $P_i \subseteq \gln$ be the $i^\th$ parabolic subgroup containing
$B$, in which the only nonzero entry outside the lower triangle may lie
at $(i,i+1)$. Consider the image $Y$ of the multiplication map $P_i
\times \ol\OO_Z \to \mn$ sending $(p,x) \mapsto p \cdot x$. This map
factors through the quotient $B\dom (P_i \times \ol\OO_Z)$ by the
diagonal action of~$B$, which is an $\ol\OO_Z$-bundle over $P_i/B \cong
\PP^1$ and hence has dimension $\dim(\ol\OO_Z) + 1$. Thus $\dim(Y) \leq
\dim(\ol\OO_Z) + 1$.
The variety $Y$ is $B \times B_+$-stable by construction. Since $B
\times B_+$ has only finitely many orbits on~$\mn$, the irreducibility
of~$Y$ implies that $\ol Y$ is an orbit closure $\ol\OO_{Z'}$ for some
partial permutation matrix~$Z'$, by Lemma~\ref{lemma:orbits}. Clearly
$\ol\OO_Z \subseteq Y$, and $s_i Z \in Y$,
% semicontinuity of $\rank(-\sub ij)$ implies that $s_i Z \not\in
% \ol\OO_Z$. Thus $\ol Y \supsetneq \ol\OO_Z$.
so the dimension bound will imply $Z' = s_i Z$ as soon as we show that $Z
\in \ol\OO_{s_i Z}$.
By
% arguments similar to those for Claim~\ref{claim:length},
checking all
% $5$ cases,
possible placements of $1$'s in rows $i$ and $i+1$ of~$s_i Z$, the
hypothesis $\length(s_i Z) < \length(Z)$ means either that $s_i Z$ has
all zeros in row~$i+1$ and a $1$ somewhere in row~$i$, or that $s_i Z$
has $1$'s at $(i,j)$ and $(i+1,j')$ with column indices $j < j'$. In the
$2 \times 2$ case, we have
\begin{eqnarray}
\nonumber
\phantom{\hbox{and}\qquad}
\left(\begin{array}{cc} 1 & 0 \\
t^{-1}& 1
\end{array}\right)
\cdot
\left(\begin{array}{cc} 1 & 0 \\
0 & 0
\end{array}\right)
\cdot
\left(\begin{array}{cc} t & 0 \\
0 &\ \ \,1\ \ \,
\end{array}\right)
&=&
\left(\begin{array}{cc} t & 0 \\
1 & 0
\end{array}\right)\phantom{.}
\\[2ex]
% \left(\begin{array}{cc}1&0\\t^{-1}&1\end{array}\right) \cdot
% \left(\begin{array}{cc}1&0\\0&1\end{array}\right)
% \cdot
% \left(\begin{array}{cc}t&1\\0&-t^{-1}\end{array}\right)
% &=& \left(\begin{array}{cc}t&1\\1&0\end{array}\right);
\label{eq:Z}
\hbox{and}\qquad
\left(\begin{array}{cc} 1 & 0 \\
t^{-1}& 1
\end{array}\right)
\cdot
\left(\begin{array}{cc} 1 & 0 \\
0 & 1
\end{array}\right)
\cdot
\left(\begin{array}{cc} t & 1 \\
0 &-t^{-1}
\end{array}\right)
&=&
\left(\begin{array}{cc} t & 1 \\
1 & 0
\end{array}\right).
\end{eqnarray}
% \begin{verbatim}
% 1 0 1 0 t 0 1 0 t 0 t 0
% = =
% t' 1 0 0 0 1 t' 0 0 1 1 0
% \end{verbatim}
View these equations as occurring in the two rows $i,i+1$ and the two
columns $j,j'$ in equations $b(t)\cdot s_i Z \cdot b_+(t)^{-1} = Z(t)$.
Inserting the appropriate extra identity rows and columns into $b(t)$ and
$b_+(t)^{-1}$ while completing the middle matrix to~$s_i Z$ yields, in
each equation, a $1$-parameter family of matrices in $\OO_Z = B s_i Z
B_+$. The limit at $t=0$ is~$Z$ in both cases, as seen from the right
hand sides.
That $\dim(\ol\OO_Z) = n^2 - \length(Z)$ follows by direct calculation
whenever $Z \subseteq \id_n$ has $1$'s only along the main diagonal.
Every other partial permutation matrix can be obtained from a diagonal
one by repeatedly switching adjacent rows. After fixing the number of
$1$'s, induction on $\length(Z)$ proves the formula.
The argument above shows that $\ol\OO_{s_i Z} = \ol Y$ is stable under
multiplication by $P_i$ on the left. In particular, $s_i \in P_i$ takes
$\ol\OO_{s_i Z}$ to itself.
\end{proof}
The next result appears in \cite{FulDegLoc}, where it is derived from the
corresponding result on flag manifolds. Again, we prove it here to keep
the exposition self-contained.
\begin{prop} \label{prop:BwB}
The matrix Schubert variety $\ol X_w$ is the closure $\ol{B w^T B_+}$ of
the \hbox{$B \times B_+$} orbit on~$\mn$ through the permutation matrix
$w^T$. Thus $\ol X_w$ is irreducible of dimension $n^2 - \length(w)$,
and $w^T$ is a smooth point of it.
\end{prop}
\begin{proof}
The stability of $\ol X_w$ under $B \times B_+$, which we explained
earlier, means that $\ol X_w$ is a union of orbits. By
Lemma~\ref{lemma:orbits} and the obvious containment $\ol\OO_{w^T}
\subseteq \ol X_w$ (sweeping down and right preserves northwest ranks),
it therefore suffices to show that any partial permutation matrix $Z$
lying in $\ol X_w$ lies also in $\ol{B w^T B_+}$.
Scaling the rows independently, we see that the $B \times B_+$ orbit
through any partial permutation matrix contains all of the so-called
``monomial matrices'' supported on~it, in which the `$1$' entries can be
replaced by arbitrary elements of~$\kk^*$. In particular, $Z$ lies in
$\ol{B w^T B_+}$ if $Z \subseteq w^T$. On the other hand, if $Z
\not\subseteq w^T$ but still $Z \in \ol X_w$, then
Lemma~\ref{lemma:bruhat} produces a permutation $v = wt_{i,i'}$ for some
$i < i'$ such that $Z \in \ol X_v$ and $\length(v) > \length(w)$. It is
enought to show that $v^T \in \ol{B w^T B_+}$.
% In the $2 \times 2$ case, we have
% \begin{eqnarray} \label{eq:BwB}
% \left(\begin{array}{cc}1&0\\t^{-1}&1\end{array}\right) \cdot
% \left(\begin{array}{cc}1&0\\0&1\end{array}\right)
% \cdot
% \left(\begin{array}{cc}t&1\\0&-t^{-1}\end{array}\right)
% &=& \left(\begin{array}{cc}t&1\\1&0\end{array}\right).
% \end{eqnarray}
View~(\ref{eq:Z}) as the two rows $i,i'$ and the two columns $w(i),w(i')$
in an equation $b(t)\cdot w^T\cdot b_+(t)^{-1} = v(t)$. Inserting the
appropriate extra identity rows and columns into $b(t)$ and $b_+(t)^{-1}$
while completing the middle matrix to~$w^T$ yields a $1$-parameter family
of matrices in $B w^T B_+$, whose limit at $t=0$ is~$v^T$.
The last sentence of the Proposition is standard for orbit closures,
except for the dimension count, which comes from Lemma~\ref{lemma:Z}.
\end{proof}
Throughout the rest of Sections~\ref{sec:intro} and~\ref{sec:gb}, we
shall repeatedly invoke the hypothesis $\length(ws_i) < \length(w)$. In
terms of permutation matrices, this means that $w^T$ differs from
$(ws_i)^T$ only in rows $i$ and $i+1$, where they look heuristically like
\begin{equation} \label{eq:w}
\begin{array}{c@{\qquad\quad}c}
% 12 3 4 5 6 7 8
\begin{array}{lr|c|c|@{}c@{}|c|c|l}
\multicolumn{6}{c}{}&\multicolumn{1}{c}{
\begin{array}{@{}c@{}}
\makebox[0pt]{$\scriptstyle w(i)$}\\
\downarrow
\end{array}}&
\multicolumn{1}{c}{\begin{array}{@{}c@{}}\\\adots\end{array}}
\\\cline{3-7}
i && & \phantom{1} & \toplinedots & \phantom{1} & 1
\\\cline{3-4}\cline{6-7}
i+1&& 1 & & & &
\\\cline{3-7}
\multicolumn{1}{c}{}&
\multicolumn{1}{c}{\begin{array}{@{}c@{}}\adots\\ \\ \end{array}}&
\multicolumn{1}{c}{
\begin{array}{@{}c@{}}
\uparrow\\
\makebox[0pt]{$\scriptstyle w(i+1)$}
\end{array}}
\\
\multicolumn{2}{c}{}&
\multicolumn{5}{c}{w^T}
\end{array}
&
% 12 3 4 5 6 7 8
\begin{array}{lr|c|c|@{}c@{}|c|c|l}
\multicolumn{2}{c}{}&
\multicolumn{1}{c}{
\begin{array}{@{}c@{}}
\makebox[0pt]{$\scriptstyle w(i+1)$}\\
\downarrow
\end{array}}&
\multicolumn{4}{c}{}&
\multicolumn{1}{c}{\begin{array}{@{}c@{}}\\\adots\end{array}}
\\\cline{3-7}
i && 1 & \phantom{1} & \toplinedots & \phantom{1} &
\\\cline{3-4}\cline{6-7}
i+1&& & & & & 1
\\\cline{3-7}
\multicolumn{1}{c}{}&
\multicolumn{1}{c}{\begin{array}{@{}c@{}}\adots\\\\\end{array}}&
\multicolumn{4}{c}{}&\multicolumn{1}{c}{
\begin{array}{@{}c@{}}
\uparrow\\
\makebox[0pt]{$\scriptstyle w(i)$}
\end{array}}
\\
\multicolumn{2}{c}{}&
\multicolumn{5}{c}{(ws_i)^T}
\end{array}
\end{array}
\end{equation}
between columns $w(i+1)$ and $w(i)$. Since reversing the inequality in
$\length(ws_i) < \length(w)$ makes so much difference, we always write
the hypothesis this way, for consistency, even though we may actually
{\em use} one of the following equivalent formulations in any given lemma
or proposition. We hope that collecting this list of standard---and at
this point almost trivially equivalent---statements (``shorter
permutation $\iff$ bigger variety'') will prevent the reader from
stumbling on this subtlety as many times as we did. In particular, the
string of characters `$\length(ws_i) < \length(w)$' can serve as a visual
cue to this frequent assumption; we shall {\em never} assume the opposite
inequality.
\begin{cor} \label{cor:bruhat}
The following are equivalent for a permutation $w \in S_n$.
% be a permutation and $s_i$ the transposition switching $i$ and~$i+1$.
$$
\begin{array}{rl@{\qquad}rl}
1.&\length(ws_i) < \length(w). &6.&\dim(\ol X_{ws_i}) > \dim(\ol X_w).
\\2.&\length(ws_i) = \length(w) - 1.&7.&\dim(\ol X_{ws_i}) = \dim(\ol X_w) + 1.
\\3.&w(i) > w(i+1). &8.&s_i \ol X_w \neq \ol X_w.
\\4.&ws_i(i) < ws_i(i+1). &9.&s_i \ol X_{ws_i} = \ol X_{ws_i}.
\\5.&I(\ol X_{ws_i}) \subset I(\ol X_w).&10.&\ol X_{ws_i} \supset \ol X_w.
\end{array}
$$
Here, the
% [commented out the following, because it can be confusing: the group
% action of $S_n$ on $\mn$ by multiplication on the left is actually a
% {\em right} group action, since we're thinking of elements of $S_n$ as
% the {\em transposes} of the corresponding permutation matrices.
% However, for just a single $s_i$, the difference disappears.]
% group $S_n$ acts on the left of $\mn$ by virtue of its containment in
% $\gln$. In particular, the
transposition $s_i$ acts on the left of $\mn$, switching rows $i$
and~$i+1$.
\end{cor}
\begin{proof}
The equivalence of $1$--$4$ comes from Claim~\ref{claim:length}. The
equivalence of these with $5$--$7$ and~$10$ uses
Proposition~\ref{prop:BwB} and Lemma~\ref{lemma:bruhat}. Finally,
$8$--$10$ are equivalent by Lemma~\ref{lemma:Z} and its proof.%
\end{proof}
\subsection{Multidegrees of matrix Schubert varieties}%%%%%%%%%%%%%%%%%%%
\label{sub:multimat}%%%%%
This subsection provides a proof of the recursion satisfied by the
multidegrees of matrix Schubert varieties, stated in
Theorem~\ref{thm:oracle}.
Let $T\times T^{-1}$ be the $2n$-dimensional torus inside $B \times B_+$.
We write $T^{-1}$ for the right factor to indicate that it acts by
inverse multiplication, and to distinguish it from the left factor.
Note, however, that $T^{-1}$ is the {\em same} torus $B \cap B_+$ as the
left factor, but with a different action. We use the left factor much
more often and call it simply~$T$.
Using the action of $T \times T^{-1}$, we can speak of $\ZZ^{2n}$-graded
multidegrees for subschemes of~$\mn$. Let $x_i : T \to \kk^*$ be the
character that picks out the $(i,i)$ entry of every diagonal matrix, so
$\{x_i\}_{i=1}^n$ constitutes the standard basis for the weight lattice
$\ZZ^n$ of~$T$. Likewise let $\{y_i\}_{i=1}^n$ be the standard basis for
the weight lattice of~$T^{-1}$. Then $\ZZ^{2n}$-graded multidegrees live
in $\ZZ[\xx,\yy] := \ZZ[x_1,\ldots,x_n,y_1,\ldots,y_n]$, with the weight
of $z_{ij}$ being $x_i - y_j$. When we speak of $\ZZ^n$-graded
multidegrees for subschemes of~$\mn$, we {\em always} mean the grading by
virtue of the $T$ action, in which $z_{ij}$ has weight~$x_i$.
\begin{example} \label{ex:w0deg}
The matrix Schubert variety $\ol X_{w_0}$ from Example~\ref{ex:w0} has
$\ZZ^{2n}$-graded multidegree $\prod_{i+j\leq n} (x_i - y_j)$. If one
only wants to consider the left half of the torus action, the
$\ZZ^n$-multidegree is $\prod_{i=1}^n x_i^{n-i}$, obtained from the
previous formula by setting $y_j$ to~$0$ for all~$j$, as per
Lemma~\ref{lemma:coarsen}.
\end{example}
\begin{example} \label{ex:s3deg}
Five of the six $3\times 3$ matrix Schubert varieties in
Example~\ref{ex:s3} have $\ZZ^{2n}$-multidegrees that are products of
expressions having the form $x_i-y_j$ as in~(\ref{eq:wt}):
\begin{eqnarray*}
[\ol X_{123}] &=& 1
\\{}
[\ol X_{213}] &=& x_1 - y_1
\\{}
[\ol X_{231}] &=& (x_1 - y_1)(x_1-y_2)
\\{}
[\ol X_{312}] &=& (x_1 - y_1)(x_2-y_1)
\\{}
[\ol X_{321}] &=& (x_1 - y_1)(x_1-y_2)(x_2-y_1)
\end{eqnarray*}
The last one, $\ol X_{132}$, has a multidegree
\begin{eqnarray*}
\ol X_{132} &=& x_1 + x_2 - y_1 - y_2 \qquad\quad\ \
\end{eqnarray*}
that can be written as a sum of expressions $(x_i-y_j)$ in two different
ways. To see why, pick term orders that choose different leading
monomials for $z_{11}z_{22} - z_{12}z_{21}$. Geometrically, these
degenerate $\ol X_{132}$ either to the scheme defined by $z_{11}z_{22}$
or to the scheme defined by $z_{12}z_{21}$, while preserving the
multidegree in both cases. The degenerate limits break up as unions
\begin{eqnarray*}
\ol X_{132}'
&=& \{Z \,|\, z_{11} = 0\} \cup \{Z\,|\, z_{22} = 0\}
\ \:=\ \:\{Z \in M_3\,|\, z_{11}z_{22} = 0\}
\\
\ol X_{132}''
&=& \{Z\,|\, z_{12} = 0\} \cup \{Z\,|\, z_{21} = 0\}
\ \:=\ \:\{Z \in M_3\,|\, z_{12}z_{21} = 0\}
\end{eqnarray*}
and therefore have multidegrees
\begin{eqnarray*}
[\ol X_{132}'] &=& (x_1-y_1) + (x_2-y_2)
\\{}
[\ol X_{132}'']&=& (x_1-y_2) + (x_2-y_1).
\end{eqnarray*}
Either way lets us calculate $[\ol X_{132}] = x_1 + x_2 - y_1 - y_2$ by
Corollary~\ref{cor:grobdeg}. For most permutations $w$, only the
antidiagonal degeneration $\ol X_{132}''$ will generalize.
\end{example}
Our goal in this section is to show that the multidegrees $\{[\ol
X_w]\}_{w \in S_n}$ of matrix Schubert varieties satisfy a recurrence
relation, with which they can be determined from the top one $[\ol
X_{w_0}]$. Define the \bem{divided difference operator} $\partial_i$
that takes a polynomial $f$ in $\{x_1,\ldots,x_n\}$ and possibly some
other variables to
\begin{eqnarray*}
\partial_i f(x_1, x_2, \ldots) &=& \frac{f(x_1, x_2, \ldots, ) - f(x_1,
\ldots, x_{i-1}, x_{i+1}, x_i, x_{i+2}, \ldots)}{x_i - x_{i+1}}.
\end{eqnarray*}
Since the numerator vanishes when $x_i=x_{i+1}$, and polynomials enjoy
unique factorization, this is again a polynomial, of degree one lower
than that of~$f$.
We will want to apply these operators to $\ZZ^n$-graded as well as
$\ZZ^{2n}$-graded multidegrees. Note that in this case, when we switch
the variables $x_i$ and $x_{i+1}$, we do {\em not\/} switch the
corresponding $\yy$ variables.
\begin{thm}\label{thm:oracle}
If the permutation $w$ satisfies $\length(ws_i) < \length(w)$, then
\begin{eqnarray*}
[\ol X_{ws_i} ] &=& \partial_i [\ol X_w]
\end{eqnarray*}
hold for both the $\ZZ^{2n}$-graded and $\ZZ^n$-graded multidegrees.
\end{thm}
Two
% more or less standard
lemmas are required before getting into the multidegree part of the
proof.
% In what follows, the transposition $t_{q,q'}$ switches rows~$q$
% and~$q'$.
\begin{lemma} \label{lemma:si}
Let $Z$ be a partial permutation matrix and $w \in S_n$.
% a permutation.
If the orbit closure $\ol\OO_Z$ has codimension~$1$ inside $\ol
X_{ws_i}$, then
% there is a transposition $t_{q,q'}$ such that $v = v't_{q,q'}$ and
% $\length(v) = \length(v') + 1$. If, in addition, $v' = ws_i$ and
% $\length(ws_i) < \length(w)$, then $v \neq w$ implies $s_i\ol X_v = \ol
% X_v$.
$\ol\OO_Z$ is mapped to itself by $s_i$ unless~$Z = w$.
\end{lemma}
\begin{proof}
First note that $\length(Z) = \length(ws_i) + 1$ by Lemma~\ref{lemma:Z}
and Proposition~\ref{prop:BwB}. Using Lemma~\ref{lemma:bruhat} and
Claim~\ref{claim:length}, we find that $Z$ is obtained from $(ws_i)^T$ by
% some sequence of
switching some pairs of rows to make partial permutations of strictly
larger length and then deleting some $1$'s. Since the length of $Z$ is
precisely one less than that of~$ws_i$, we can switch exactly one pair of
rows of~$(ws_i)^T$, or we can delete a single~$1$ from $(ws_i)^T$.
Any $1$ that we delete from $(ws_i)^T$ must have no $1$'s southeast of
it, or else the length increases by more than one, as direct calculation
shows. Therefore the~$1$ in row~$i$ of $(ws_i)^T$ cannot be deleted by
part~4 of Corollary~\ref{cor:bruhat}, leaving us in the situation of
Lemma~\ref{lemma:Z} with $Z = w^T$ (see~(\ref{eq:Z}) and the sentence
before it).
% For the second, first note that $s_i\ol X_v = \ol X_v$ if and only if
% $v(i) < v(i+1)$, by $4 \iff 9$ in Corollary~\ref{cor:bruhat} (with
% $ws_i = v$).
The $1$'s in any pair of switched rows must enclose no additional $1$'s
in their rectangle, by Claim~\ref{claim:length}. Suppose now that $v
\neq w$, but that switching rows $q$ and~$q'$ of $(ws_i)^T$ results in a
permutation matrix $v^T$ for which $v(i) > v(i+1)$. Then exactly one of
$q$ and~$q'$ must lie in $\{i,i+1\}$, because switching rows $i$
and~$i+1$ yields $v = w$, but moving neither leaves $v(i) < v(i+1)$.
Either the $1$ at $(i,w(i+1))$ or the $1$ at $(i+1,w(i))$ lies inside the
rectangle formed by the switched $1$'s.
% , so $\length(v) \leq \length(ws_i) - 3$ by Claim~\ref{claim:length}.
\end{proof}
\begin{lemma} \label{lemma:regular}
If $\length(ws_i) < \length(w)$, and $\mm_{ws_i}$ is the maximal ideal in
the local ring of $(ws_i)^T \in \ol X_{ws_i}$, then the variable
$z_{i+1,w(i+1)}$ maps to a regular parameter in~$\mm_{ws_i}$. In other
words $z_{i+1,w(i+1)}$ lies in $\mm_{ws_i} \minus \mm_{ws_i}^2$.
\end{lemma}
\begin{proof}
Let $v = ws_i$, and consider the map $B \times B_+ \to \mn$ sending
$(b,b^+) \mapsto b \cdot v^T \cdot b^+$. The image of this map is
contained in $\ol X_v$ by Proposition~\ref{prop:BwB}, and the identity
$\id := (\id_B,\id_{B_+})$ maps to $v^T$. The induced map of local rings
the other way therefore takes $\mm_v$ to the maximal ideal
\begin{eqnarray*}
\mm_\id &:=& \
+ \ j\>
\end{eqnarray*}
in the local ring at the identity $\id \in B \times B_+$. It is enough
to demonstrate that the image of $z_{i+1,w(i+1)}$ lies in $\mm_\id \minus
\mm_\id^2$.
Direct calculation shows that $z_{i+1,w(i+1)}$ maps to
\begin{eqnarray*}
b_{i+1,i}b^+{}_{\!\!\!\!\!w(i+1),w(i+1)} + \sum_{q \in Q}
b_{i+1,q}b^+{}_{\!\!\!\!\!q,w(i+1)} \quad\hbox{where}\quad Q = \{q < i
\mid w(q) < w(i+1)\}.
\end{eqnarray*}
In particular, all of the summands $b_{i+1,q}b^+{}_{\!\!\!\!\!q,w(i+1)}$
lie in $\mm_\id^2$. On the other hand, $b^+{}_{\!\!\!\!\!w(i+1),w(i+1)}$
is a unit near the identity, so
$b_{i+1,i}b^+{}_{\!\!\!\!\!w(i+1),w(i+1)}$ lies in $\mm_\id \minus
\mm_\id^2$.
\end{proof}
The argument forming the proof of the previous lemma is an alternative
way to calculate the dimension as in Lemma~\ref{lemma:Z}.
\begin{proofof}{Theorem~\ref{thm:oracle}}
The proof here works for the $\ZZ^n$-grading as well as the
$\ZZ^{2n}$-grading, simply by ignoring all occurrences of $\yy$, or
setting them to zero.
Let $j = w(i+1) - 1$, and suppose $\rank(w^T\sub ij) = r-1$. Then the
permutation matrix $(ws_i)^T$ has $r$ entries equal to $1$ in the
submatrix $(ws_i)^T\sub ij$. Consider the $r \times r$ minor $\Delta$
using the rows and columns in which $(ws_i)^T\sub ij$ has $1$'s. Thus
$\Delta$ is not the zero function on $\ol X_{ws_i}$; in fact, $\Delta$ is
nonzero everywhere on its interior $B (ws_i)^T B_+$. Therefore
% , up to components of codimension at least $2$ in~$\ol X_{ws_i}$,
the subscheme $X_\Delta$ defined by $\Delta$ inside $\ol X_{ws_i}$ is
supported on a union of orbit closures $\ol\OO_Z$
% matrix Schubert varieties
contained in $\ol X_{ws_i}$ with codimension~$1$.
Compare the subscheme $X_\Delta$ to its image $s_i X_\Delta = X_{s_i
\Delta}$ under switching rows~$i$ and~$i+1$. Lemma~\ref{lemma:si} says
that $s_i$ induces an automorphism of the local ring at the generic point
(i.e.\ the prime ideal) of $\ol\OO_Z$ inside $\ol X_{ws_i}$, for every
irreducible component $\ol\OO_Z$ of $X_\Delta$ other than~$\ol X_w$.
This automorphism takes $\Delta$ to $s_i\Delta$, so these two functions
have the same multiplicity along $\ol\OO_Z$. The only remaining
codimension~$1$ irreducible component of $X_\Delta$ is $\ol X_w$, and we
shall now verify that the multiplicity equals~$1$ there. As a
consequence, the multiplicity of $s_i X_\Delta$ along $s_i \ol X_w$ also
equals~$1$.
% {bf multiplicity of $\Delta$ along $\pp$ equals multiplicity of $s_i
% \Delta$ along $s_i \pp$}
By Proposition~\ref{prop:BwB}, the local ring of $(ws_i)^T$ in $\ol
X_{ws_i}$ is regular. Since $s_i$ is an automorphism of $\ol X_{ws_i}$,
we find that the local ring of $w^T \in \ol X_{ws_i}$ is also regular.
In a neighborhood of $w^T$, the variables $z_{qp}$ corresponding to the
locations of the $1$'s in $w^T\sub ij$ are units. This implies that the
coefficient of $z_{i,w(i+1)}$ in $\Delta$ is a unit in the local ring of
$w^T \in \ol X_{ws_i}$. On the other hand, the set of variables in spots
where $w^T$ has zeros generate the maximal ideal in the local ring at
$w^T \in \ol X_{ws_i}$. Therefore, all of the terms of $\Delta$ lie in
the square of this maximal ideal, except for the unit times
$z_{i,w(i+1)}$ term produced above. Hence, to prove multiplicity one, it
is enough to prove that $z_{i,w(i+1)}$ itself is a regular parameter at
$w^T \in \ol X_{ws_i}$, or equivalently (after applying $s_i$) that
$z_{i+1,w(i+1)}$ is a regular parameter at $(ws_i)^T \in \ol X_{ws_i}$.
This is Lemma~\ref{lemma:regular}.
Now we come to the multidegree trick. Consider $\Delta$ and $s_i\Delta$
not as elements in $\kk[\zz]$, but as elements in the ring $\kk[\zz][u]$
from Section~\ref{sub:alex}, whose spectrum we denote by $\mn \times
\AA^1$. Then $\Delta$ and the product $u s_i\Delta$ in $\kk[\zz][u]$
have the {\em same} (ordinary) weight $f := f(\xx,\yy)$. Since the
affine coordinate ring of $\ol X_{ws_i}$ is a domain, and neither
$\Delta$ nor $s_i\Delta$ vanishes on $\ol X_{ws_i}$, we get two short
exact sequences
\begin{equation} \label{eq:Q}
0\ \to\ \kk[\zz][u]/I(\ol X_{ws_i})\kk[\zz][u](-f)\
\stackrel\Theta\too\ \kk[\zz][u]/I(\ol X_{ws_i})\kk[\zz][u]\ \too\
Q(\Theta)\ \to\ 0,
\end{equation}
in which $\Theta$ equals either $\Delta$ or $us_i\Delta$. The quotients
$Q(\Delta)$ and $Q(us_i\Delta)$ therefore have equal $\ZZ^{2n}$-graded
Hilbert series, and hence equal multidegrees.
Note that $Q(\Delta)$ is the coordinate ring of $X_\Delta \times \AA^1$,
while $Q(us_i\Delta)$ is the coordinate ring of $(X_\Delta \times \AA^1)
\cup (\ol X_{ws_i} \times \{0\})$, the latter component being the zero
scheme of $u$ in $\kk[\zz][u]/I(\ol X_{ws_i})\kk[\zz][u]$. Breaking up
the multidegrees of $Q(\Delta)$ and $Q(us_i\Delta)$ into sums over
irreducible components as in Proposition~\ref{prop:additive'}, the
analysis of multiplicity above says that almost all the terms in the
equation
\begin{eqnarray*}
[X_\Delta \times \AA^1] &=& [s_i X_\Delta \times \AA^1] + [\ol X_{ws_i}
\times \{0\}]
\end{eqnarray*}
cancel, leaving us only with
\begin{eqnarray} \label{eq:cancel}
[\ol X_w \times \AA^1] &=& [s_i\ol X_w \times \AA^1] + [\ol X_{ws_i}
\times \{0\}].
\end{eqnarray}
The brackets in these equations denote multidegrees over $\kk[\zz][u]$.
However, the ideals in $\kk[\zz][u]$ of $\ol X_w \times \AA^1$ and
$s_i\ol X_w \times \AA^1$ are extended from the ideals in $\kk[\zz]$ of
$\ol X_w$ and $s_i \ol X_w$. Therefore they have the same Hilbert
numerators, whence $[\ol X_w \times \AA^1] = [\ol X_w]$ and $[s_i\ol X_w
\times \AA^1] = [s_i\ol X_w]$ as polynomials in $\xx$ and~$\yy$. The
same argument shows that $[\ol X_{ws_i} \times \AA^1] = [\ol X_{ws_i}]$.
The coordinate ring of $[\ol X_{ws_i} \times \{0\}]$, on the other hand,
is the right hand term of the exact sequence that results after replacing
$f$ by $x_i - x_{i+1}$ and $\Theta$ by $u$ in~(\ref{eq:Q}). We therefore
find that
\begin{eqnarray*}
[\ol X_{ws_i} \times \{0\}] &=& (x_i - x_{i+1})[\ol X_{ws_i} \times
\AA^1]\ \:=\ \:(x_i - x_{i+1})[\ol X_{ws_i}]
\end{eqnarray*}
as polynomials in $\xx$ and $\yy$. Substituting back
into~(\ref{eq:cancel}) yields the equation $[\ol X_w] = [s_i\ol X_w] +
(x_i-x_{i+1})[\ol X_{ws_i}]$, which produces the desired result after
moving the $[s_i\ol X_w]$ to the left and dividing through by $x_i -
x_{i+1}$.
\end{proofof}
\begin{remark}
This proof, although translated into the language of multigraded
commutative algebra, is actually derived from a standard proof of divided
difference formulae by localization in equivariant cohomology, when $\kk
= \CC$. To see how, our two functions $\Delta$ and $s_i\Delta$ yield a
map $\ol X_{ws_i} \to \CC^2$, where the preimage of one axis is $\ol X_w$
union some stuff, and the preimage of the other axis is $s_i \ol X_w$
union the same stuff. Therefore all of the unwanted (canceling)
contributions map to the point $(0,0) \in \CC^2$. Essentially, the
standard equivariant localization proof makes the map to $\CC^2$ into a
map to $\CC\PP^1$, thus avoiding the extra components, and pulls back the
localization formula on $\CC\PP^1$ to a formula on whatever $\ol
X_{ws_i}$ has become (a Schubert variety).
\end{remark}
If $w \in S_n$ does not have a descent at~$i$, so $w(i) > w(i+1)$, then
the result of applying the divided difference operator is less
interesting:
\begin{eqnarray*}
\partial_i [\ol X_w] &=& \partial_i \partial_i [\ol X_{ws_i}]\ \:=\ \:0
\end{eqnarray*}
because $\partial_i f$ is symmetric in $x_i$ and $x_{i+1}$ for any
polynomial $f$, and thus $\partial_i^2 f$ is zero.
\subsection{Schubert and Grothendieck polynomials}\label{sub:schubintro}%
The polynomials appearing in Theorem~\ref{thm:oracle} are our title
characters:
\begin{defn} \label{defn:schubert}
The \bem{Schubert polynomial} and \bem{double Schubert polynomial} for a
permutation $w \in S_n$ are the $\ZZ^n$-graded and $\ZZ^{2n}$-graded
multidegrees
\begin{eqnarray*}
\SS_w(\xx)\ :=\ [\ol X_w]_{\ZZ^n} &\quad\hbox{ and }\quad&
\SS_w(\xx,\yy)\ :=\ [\ol X_w]_{\ZZ^{2n}}.
\end{eqnarray*}
\end{defn}
The definition says that $w \in S_n$, but in fact $n$ plays no role,
because the multidegrees $[\ol X_w]$ are stable under the inclusion $S_n
\into S_{n+1}$, in the following sense. Given a permutation $w_n$ in
$S_n$, let $w_{n+1}$ be its extension to a permutation in $S_{n+1}$
defined by fixing the last element,~\hbox{$n+1$}. Then $\ol X_{w_{n+1}}$
is the set of $(n+1) \times (n+1)$ matrices where the upper left $n\times
n$ block is in $\ol X_{w_n}$, while the last row and column are arbitrary
(and so drop out of the multidegree calculation). Thus $[\ol X_{w_n}] =
[\ol X_{w_{n+1}}]$.
Lascoux and Sch\"utzenberger \cite{LSpolySchub} used the recursion in
Theorem~\ref{thm:oracle} as the definition of Schubert polynomials and
double Schubert polynomials, and they proved the stability property,
which singles $\SS_w$ out uniquely among polynomials representing the
cohomology classes of the Schubert variety corresponding to~$w$ in the
flag manifold (see Appendix~\ref{app:basics}). To have it on record, we
state their definition as a corollary.
\begin{cor} \label{notdefn:schubert}
Set $\SS_{w_0}(\xx) = \prod_{i=1}^n x_i^{n-i}$ and $\SS_{w_0}(\xx,\yy) =
\prod_{i+j \leq n} (x_i-y_j)$. The Schubert and double Schubert
polynomials for $w$ are $\SS_w(\xx) = \partial_{i_k} \cdots
\partial_{i_1} \SS_{w_0}(\xx)$ and $\SS_w(\xx,\yy) = \partial_{i_k}
\cdots \partial_{i_1} \SS_{w_0}(\xx,\yy)$, where $w_0w = s_{i_1} \cdots
s_{i_k}$ and\/ $\length(w_0w)=k$.\hfill$\Box$
\end{cor}
The condition $\length(w_0w)=k$ means by definition that $w_0w = s_{i_1}
\cdots s_{i_k}$ is a \bem{reduced expression} for $w_0 w$. The
formulation in Corollary~\ref{notdefn:schubert} seems to make it less
clear that $\SS_w$ is well-defined: it is necessary that the divided
differences satisfy the \bem{Coxeter relations}, $\partial_i
\partial_{i+1} \partial_i = \partial_{i+1} \partial_i \partial_{i+1}$ and
$\partial_i \partial_{i'} = \partial_{i'} \partial_i$ when $|i-i'| \geq
2$. They do, of course (by Theorem~\ref{thm:oracle}, for the cases we
care about), and this is not hard to check directly.
As the ordinary Schubert polynomials are much more common in the
literature than double Schubert polynomials, we have phrased much of our
coming exposition in terms of Schubert polynomials (left torus actions,
or $\ZZ^n$-graded multidegrees), although the double Schubert version
(with the $T \times T^{-1}$ action, or $\ZZ^{2n}$-graded multidegrees)
holds with the same proof, mutatis mutandis. This choice has the
advantage of simplifying the notation.
\begin{example}
% {bf I'd like a diagram here, with $[\ol X_{321}] = x_1^2 x_2$ at the
% top, downward edges labeled by $\partial_1$ and $\partial_2$, half of
% which lead to $0$s.}
% you want your diagram? here's your diagram...
Here are all of the Schubert polynomials for permutations in $S_3$, along
with the rules for applying divided differences.
$$
\begin{array}{@{}r@{\hspace{-1ex}}r@{}r@{}c@{}l@{}l@{\hspace{-1ex}}l@{}}
& & &[\ol X_{321}]& & &
\\ & & \ld52\swarrow& &\searrow\rd51 & &
\\ & & [\ol X_{312}]& &[\ol X_{231}] & &
\\&\ld52\swarrow&\downarrow\ \rd21\ &&\ \ld22\ \downarrow&\searrow\rd51
\\ 0 & & [\ol X_{132}]& &[\ol X_{213}] & & 0
\\&\swarrow\rd01\!\!& \ld02\searrow& &\swarrow\rd01 &\ld02\searrow&
\\0\ & & &[\ol X_{123}]& & &\ \ 0
\end{array}
\qquad\qquad
\begin{array}{@{}r@{\hspace{-1ex}}r@{}r@{}c@{}l@{}l@{\hspace{-1ex}}l@{}}
& & & x_1^2x_2 & & &
\\ & & \ld52\swarrow& &\searrow\rd51 & &
\\ & &x_1^2\ \ \ \ & & \mcc{x_1x_2} & &
\\&\ld52\swarrow\!\!&\mcc{\downarrow\;\rd21}&&\mcc{\ld22\;\downarrow}&\searrow\rd51
\\ 0 & & x_1+x_2 & &\ \ \ x_1 & & 0
\\&\swarrow\rd01\!\!\!\!& \ld02\searrow& &\swarrow\rd01 &\!\!\ld02\searrow&
\\0\: & & & 1 & & &\ 0
\end{array}
$$
For a taste of what's to come, compare these diagrams to the one in
Example~\ref{ex:rc3}.%
\end{example}
Note that the use of multidegrees instead of Hilbert series greatly
simplified life in the proof of Theorem~\ref{thm:oracle}, by virtue of
their additivity on irreducible components
(Proposition~\ref{prop:additive'}). Of course, Schubert polynomials are
therefore only the leading terms of a richer structure, coming from the
Hilbert numerators themselves. In view of Theorem~\ref{thm:oracle}, it
is natural to ask whether the Hilbert numerators of matrix Schubert
varieties satisfy a similar recurrence. They do, as we shall see in
Theorem~\ref{thm:gb}. The recurrence uses a ``homogenized'' operator
(sometimes called an \bem{isobaric divided difference operator}):
\begin{defn} \label{defn:dem}
Let $R$ be a unique factorization domain. The \bem{$i^\th$ Demazure
operator} $\dem i: R[[\xx]] \to R[[\xx]]$ sends a power series $f(\xx)$
to
$$
\frac{x_{i+1}f(x_1, \ldots, x_n) - x_if(x_1, \ldots, x_{i-1}, x_{i+1},
x_i, x_{i+2}, \ldots, x_n)}{x_{i+1} - x_i}.
$$
\end{defn}
\begin{defn} \label{defn:groth}
The \bem{Grothendieck polynomial} $\GG_w(\xx)$ is defined recursively
from the top one $\GG_{w_0}(\xx) := \prod_{i=1}^n (1-x_i)^{n-i}$, and the
recurrence
\begin{eqnarray*}
\GG_{ws_i}(\xx) &=& \dem i \GG_w(\xx)
\end{eqnarray*}
whenever $\length(ws_i) < \length(w)$. The \bem{double Grothendieck
polynomials} are defined by the same recurrence, but start from
$\GG_{w_0}(\xx,\yy) := \prod_{i+j \leq n} (1-x_i y_j^{-1})$.
\end{defn}
% \begin{remark} \label{rk:groth}
It will follow from our inductive method of proof in Section~\ref{sec:gb}
that Grothendieck polynomials turn out to be well-defined; however, as
with Schubert polynomials, one can check directly that the $\dem i$
satisfy the Coxeter relations. Grothendieck polynomials enjoy the same
stability property as do Schubert polynomials, but again, this will
follow immediately from Theorem~\ref{thm:gb}.
Lascoux and Sch\"utzenberger \cite{LSgrothVarDrap}, building on ideas of
Demazure \cite{Dem} and Bernstein-Gel$'$fand-Gel$'$fand \cite{BGG},
showed that Grothendieck polynomials represent the $K$-classes of the
structure sheaves of Schubert varieties in the flag manifold. Their
methods require some vanishing of sheaf cohomology, namely the
rationality of singularities for Schubert varieties. Working in reverse,
we shall also prove that the $K$-classes of matrix Schubert varieties
satisfy this recurrence (Corollary \ref{cor:groth}), but indirectly via
multidegrees, which are easier to deal with geometrically (as seen in the
proof of Theorem~\ref{thm:oracle}). We then exploit a key property of
squarefree monomial ideals (Lemma~\ref{lemma:IN}) to draw conclusions
about $K$-theory
% (i.e.\ Hilbert series)
upon Gr\"obner deformation.
% {\bf if we get rationality of singularities, mention it here}
% \end{remark}
It is automatic that our multidegree-defined Schubert polynomials $\SS_w$
relate to the recursively-defined Grothendieck polynomials as in the next
lemma, which gives hope for a Hilbert numerator statement.
\begin{lemma} \label{lemma:schubert}
The Schubert polynomial $\SS_w(\xx)$ is the sum of all lowest-degree
terms in $\GG_w(\1 - \xx)$. Similarly, the double Schubert polynomial
$\SS_w(\xx,\yy)$ is the sum of all lowest-degree terms in
$\GG_w(\1-\xx,\1-\yy)$.
\end{lemma}
\begin{proof}
Assuming $f(\1-\xx)$ is homogeneous, plugging $\1-\xx$ for $\xx$ into the
displayed equation in Definition~\ref{defn:dem} and taking the lowest
degree terms yields $\partial_i f(\1-\xx)$. Since $\SS_{w_0}$ is
homogeneous, the result follows by induction on $\length(w_0w)$.
\end{proof}
Although the Demazure operators are usually applied only to polynomials
in~$\xx$, it will be crucial in our applications to use them on power
series in~$\xx$. Note that $\dem i$ takes a power series to a power
series, again because $R[[\xx]]$ has unique factorization. Since the
standard denominator $f(\xx) = \prod_{i=1}^n(1-x_i)^n$ for $\ZZ^n$-graded
Hilbert series over $\kk[\zz]$ is symmetric in $x_1,\ldots,x_n$, an easy
check shows that applying $\dem i$ to a Hilbert series $g/f$ simplifies:
$\dem i(g/f) = (\dem i g)/f$. The same comment applies when $f(\xx) =
\prod_{i,j=1}^n(1-x_i/y_j)$ is the standard denominator for
$\ZZ^{2n}$-graded Hilbert series.
\begin{remark} \label{rk:conventions}
We consciously chose our notational conventions (with considerable
effort) to mesh with those of \cite{FulDegLoc}, \cite{LSpolySchub},
\cite{FKgrothYangBax}, \cite{HerzogTrung}, and \cite{BB} concerning
permutations ($w^T$ versus~$w$), the indexing on (matrix) Schubert
varieties and polynomials (open orbit corresponds to identity permutation
and smallest orbit corresponds to long word), the placement of one-sided
ladders (in the northwest corner as opposed to the southwest), and
rc-graphs. These conventions dictated our seemingly idiosyncratic
choices of Borel subgroups as well as the identification $\FL \cong B
\dom \gln$ as the set of right cosets, and resulted in our use of row
vectors in $\kk^n$ instead of the usual column vectors. That there even
existed consistent conventions came as a relief to us; that they remained
consistent when combined with the signs and weights that enter into
computations in equivariant cohomology and $K$-theory, which have their
own natural bases, was more than we could have requested.
\end{remark}
%\end{section}{Introduction}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Gr\"obner bases}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:gb}
\subsection{The Gr\"obner basis theorem}\label{sub:gb}%%%%%%%%%%%%%%%%%%%
Recall our notation: $\mn$ is the $n \times n$ matrices over~$\kk$, with
coordinate ring $\kk[\zz]$ in indeterminates $\{z_{ij}\}_{i,j=1}^n$, the
northwest $q \times p$ submatrix of a matrix $Z$ is $Z\sub qp$, and the
matrix Schubert variety $\ol X_w \subseteq \mn$ consists of the matrices
$Z \in \mn$ such that $\rank(Z\sub qp) \leq \rank(w^T\sub qp)$ for all
$q,p$. We now define two associated ideals.
\begin{defn}{} \label{defn:Iw}
%\begin{defnlabeled}{{\cite[Section~3]{FulDegLoc}}}
Let $w \in S_n$ be a permutation.
\begin{thmlist}
\item
The \bem{Schubert determinantal ideal $I_w \subset \kk[\zz]$} is
generated by all minors in $Z\sub qp$ of size $1 + \rank(w^T\sub qp)$ for
all $q,p$, where $Z = (z_{ij})$ is the matrix of variables.
\item
The \bem{antidiagonal ideal} $J_w$ is generated by the antidiagonals of
the minors of $Z = (z_{ij})$ generating $I_w$.
\end{thmlist}
Here, the \bem{antidiagonal} of a square matrix or a minor is the product
of the entries on the main antidiagonal.
\end{defn}
It is clear from the definition that $\ol X_w$ is the reduced subvariety
of $\mn$ underlying the subscheme defined by $I_w$. It is equally clear
that given any \bem{antidiagonal term order~$>$}, which by definition
picks off from each minor its antidiagonal term, $J_w$ is contained in
the initial ideal $\IN_>(I_w)$ of $I_w$. There are lots of antidiagonal
term orders, including: the reverse lexicographic term order that snakes
its way from the northwest corner to the southeast corner, $z_{11} >
z_{12} > \cdots > z_{1n} > z_{21} > \cdots > z_{nn}$; or the
lexicographic term order snaking its way from northeast to southwest.
% $z_{1\,n} > z_{1\,n-1} > \cdots > z_{1\,1} > z_{2\,n} > \cdots >
% z_{n\,1}$.
That these term orders exist (Example~\ref{ex:antidiag}) ensures that our
main result isn't vacuous.
The ring $\kk[\zz]$ is both $\ZZ^n$-graded and $\ZZ^{2n}$-graded, with
the exponential weight of $z_{ij}$ being respectively $x_i$ or $x_i/y_j$,
where $\xx = x_1,\ldots,x_n$ and $\yy = y_1,\ldots,y_n$ are standard
bases for copies of $\ZZ^n$. (Other gradings will occur later on;
background on gradings for the coordinate ring of~$\mn$ can be found in
Section~\ref{sub:alex} or Appendix~\ref{app:K-theory}, especially
Example~\ref{ex:grading} and Example~\ref{ex:functorial}.) Observe that
$I_w$, $J_w$, and the radical ideal $I(\ol X_w)$ of the matrix Schubert
variety $\ol X_w$ are all $\ZZ^{2n}$-graded (and hence also
$\ZZ^n$-graded), the first two by checking the algebra and the third
because it is the radical of the $\ZZ^{2n}$-graded ideal $I_w$ (see
Lemma~\ref{lemma:grading} for a geometric explanation). We denote the
$\ZZ^n$-graded Hilbert series of a $\ZZ^n$-graded module $\Gamma$ over
$\kk[\zz]$ by $H(\Gamma; \xx)$, and write $H(\Gamma;\xx,\yy)$ for the
$\ZZ^{2n}$-graded Hilbert series if $\Gamma$ is $\ZZ^{2n}$-graded.
\begin{thm} \label{thm:gb}
If $>$ is an antidiagonal term order, then $\IN_>(I_w) = J_w$; in other
words, the $(\rank(w^T\sub qp)+1)$-minors of $Z\sub qp$ for all $q,p$
constitute a Gr\"obner basis. Furthermore, $I_w = I(\ol X_w)$, and the
variously graded Hilbert series of $\ci w$ are
\begin{eqnarray} \label{eq:groth}
H(\ci w; \xx)\:=\:\frac{\GG_w(\xx)}{\prod_{i=1}^n(1-x_i)^n}
\:&{\rm and}&\:
H(\ci w;\xx,\yy)\:=\:\frac{\GG_w(\xx,\yy)}{\prod_{i,j=1}^n(1-x_i/y_j)},
\qquad
\end{eqnarray}
the numerators being Grothendieck and double Grothendieck polynomials
for~$w$.
\end{thm}
%As we remarked in Section~\ref{sub:schubintro}, the $\ZZ^{2n}$-graded
%version of this theorem also holds, although this is the last time we
%mention it in this section.
\begin{remark}
Fulton proved that $I_w$ is the radical ideal of $\ol X_w$ in
\cite{FulDegLoc}, where he also related the ideals $I_w$ to Schubert
varieties. In Sections~\ref{sub:minimal}--\ref{sub:other}, we compare
his point of view to ours, as well as to statements about ladder
determinantal ideals (which we define later) by other authors. In the
course of our comparisons, Proposition~\ref{prop:ess} will pinpoint those
minors forming a {\em minimal\/} Gr\"ob\-ner basis.
% can we prove that the minimal Gr\"obner basis minimally generates
% $I_w$?
\end{remark}
The proof of Theorem~\ref{thm:gb}, which involves a great deal of
tailor-made combinatorics, spans the entirety of Section~\ref{sec:gb},
although we encounter some other results of independent interest along
the way. In the rest of this subsection we provide a brief argument that
constitutes a complete proof of Theorem~\ref{thm:gb}, if one assumes the
results it quotes from later subsections. Besides providing the logical
framework, the argument serves as an overview of the rest of
Section~\ref{sec:gb}.
Before getting to the proof, we present an example (that will recur a few
times).
\begin{example} \label{ex:intro}
Let $w = 13865742$, so that $w^T$ is given by the left matrix below.
\begin{rcgraph}
%\hbox{\normalsize $w^T = $ \ }
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline * & & & & & & &
\\\hline & & * & & & & &
\\\hline & & & & & & & *
\\\hline & & & & & * & &
\\\hline & & & & * & & &
\\\hline & & & & & & * &
\\\hline & & & * & & & &
\\\hline & * & & & & & &
\\\hline
\end{array}
\ \ \implies\ \
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1
\\\hline 1 & 1 & & & & & &
\\\hline 1 & 1 & & & & & &
\\\hline 1 & 1 & & & & & &
\\\hline 1 & 1 & & & & & &
\\\hline 1 & 1 & & & & & &
\\\hline 1 & 1 & & & & & &
\\\hline 1 & & & & & & &
\\\hline
\end{array}
\ ,\
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2
\\\hline 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2
\\\hline 2 & 2 & 2 & 2 & 2 & 2 & 2 &
\\\hline 2 & 2 & 2 & 2 & 2 & & &
\\\hline 2 & 2 & 2 & 2 & & & &
\\\hline 2 & 2 & 2 & 2 & & & &
\\\hline 2 & 2 & 2 & & & & &
\\\hline 2 & 2 & & & & & &
\\\hline
\end{array}
\end{rcgraph}
Then each matrix in $\ol X_w \subseteq \mn$ has the property that every
rectangular submatrix contained in the region filled with $1$'s has rank
$\leq 1$, and every rectangular submatrix contained in the region filled
with $2$'s has rank $\leq 2$. The ideal $I_w$ contains the $21$
% ${7 \choose 6}$
minors of size $2 \times 2$ in the first region and the $144$
%
\begin{comment}%
{$$
\textstyle
[{7 \choose 3} \cdot 1 + {6 \choose 3} \cdot {4 \choose 3} + {4 \choose
3} \cdot {5 \choose 3} + 1 \cdot {7 \choose 3}] - [{6 \choose 3} \cdot
1 + {4 \choose 3} \cdot {4 \choose 3} + 1 \cdot {5 \choose 3}]
$$
$$
\textstyle
[35 + 20 \cdot 4 + 4 \cdot 10 + 35] - [20 + 4 \cdot 4 + 10] =
[35 + 80 + 40 + 35] - [20 + 16 + 10] = 190 - 46 = 144
$$
$$
\textstyle
[{7 \choose 3}] + [{6 \choose 3} \cdot {3 \choose 2}] + [{4 \choose 3}
\cdot {4 \choose 2}] + [2 \cdot 1 \cdot {5 \choose 2} + 1 \cdot 1
\cdot {5 \choose 1}]
$$
$$
\textstyle
[35] + [20 \cdot 3] + [4 \cdot 6] + [20 + 5] = 35 + 60 + 24 + 25 = 144
$$}%
\end{comment}%
%
minors of size $3 \times 3$ in the second region. Moreover, it is easy
to show directly that all of the larger minors in $I_w$ stipulated by the
definition have antidiagonals divisible by the antidiagonals of some $2$-
or $3$-minor as above (this will also follow from
Proposition~\ref{prop:ess}, where we describe explicitly the minimal
Gr\"obner basis). Therefore, the $165$ minors of size $2 \times 2$ and
$3 \times 3$ in $I_w$ form a Gr\"obner basis for $I_w$.
\end{example}
% {bf for the following argument, we need multidegrees to give us
% ordinary degrees, which means we need that contained-in-half-space
% thing. I would rather build it into the definition of multigraded, back
% in the multidegrees intro, since it's all WE need. Then mention that
% it's generalizable by those people who care.}
%
% {bf don't need multidegree to give us ordinary degrees; need all
% coordinate subspaces to have nonzero multidegree---that's it, because
% then we know that setting all $\xx$ variables to 1 yields the number of
% subspaces.}
In keeping with the convention of this monograph to state results in the
main body only for the gradings we're considering, we assume the ideals
$I$ and $J$ in what follows are $\ZZ^n$-graded in the coordinate ring
of~$\mn$. However, the result holds for any multigrading on any
polynomial ring that refines the usual $\ZZ$-grading.
% in which all of the irreducible components of the zero set of $J$ have
% nonzero multidegrees.
\begin{lemma} \label{lemma:IN}
Suppose $I$ is a multigraded ideal and $J$ is an equidimensional
squarefree monomial ideal contained in the initial ideal $\IN(I)$ of $I$
for some term order. If the zero sets of $I$ and $J$ have the same
dimension and their multidegrees $\cC(\kk[\zz]/I;\xx) =
\cC(\kk[\zz]/J;\xx)$ agree, then $\IN(I) = J$, and $I$ is reduced.
\end{lemma}
\begin{proof}
% Multidegree is automatically preserved under taking initial ideals {bf
% by definition, right?}, since the Hilbert series is preserved under
% flat deformation. Fix a $\ZZ$-grading coarsening our multigrading, as
% the rest of the argument will only use a single grading, and the
% multidegree reduces to the ordinary degree.
By Proposition~\ref{prop:degree}, the multidegree of $\kk[\zz]/J$ is a
sum of monic monomials, one for each irreducible component in its zero
set. Since $J \subseteq \IN(I)$, any maximal dimensional irreducible
component of the scheme defined by $\IN(I)$ is contained in one of these
subspaces, and hence equal to it (and thus reduced) by comparing
dimensions. Proposition~\ref{prop:additive'} says that the multidegree
of $\kk[\zz]/\IN(I)$ is therefore also a sum of monomials, one for each
irreducible component of $J$ that happens also to be an irreducible
component of $\IN(I)$. But the first sentence of
Corollary~\ref{cor:grobdeg} with $\yy = \0$ says that the multidegree of
$\kk[\zz]/\IN(I)$ equals that of $\kk[\zz]/I$. By hypothesis, the
multidegrees of $\kk[\zz]/\IN(I)$ and $\kk[\zz]/J$ coincide, and we
conclude that $\IN(I) \subseteq J$.
Since $I$ degenerates to $J$, and $J$ (being squarefree) is reduced, $I$
is reduced too.%
\end{proof}
The $\ZZ$-graded version of this lemma, along with the ensuing conclusion
that a candidate Gr\"obner basis actually is one, appears also in
\cite{MartinPhD}, for a different ideal.
\begin{proofof}{Theorem~\ref{thm:gb} [using the rest of
Section~\ref{sec:gb}]}
%
The hardest step in the proof is Corollary \ref{cor:induction}, which
shows that the Hilbert series for $\{\cj w\}_{w \in S_n}$ satisfy the
Demazure recursion defining the Grothendieck polynomials. The Hilbert
numerator $\KK(\cj {w_0};\xx)$ is easily seen to be $\GG_{w_0}(\xx)$, so
Corollary~\ref{cor:induction} and downward induction on $\length(w)$
imply that the Hilbert numerator of $\cj w$ is $\GG_w(\xx)$. Therefore
the Hilbert series of $\cj w$ matches the right hand side
of~(\ref{eq:groth}).
Lemma \ref{lemma:schubert} immediately implies that the multidegree of
$\cj w$ is the Schubert polynomial $\SS_w(\xx)$, which agrees with the
multidegree of the corresponding matrix Schubert variety by
Theorem~\ref{thm:oracle}.
Next we show that $J_w$ is equidimensional of dimension $\dim \ol X_w$.
This is completed in Proposition~\ref{prop:pure}, which applies some of
the combinatorics arising in the calculation of the Hilbert series of
$\cj w$ to the facets of its Stanley--Reisner simplicial complex.
At this point we can apply our Lemma \ref{lemma:IN}, concluding that
$\IN(I_w) = J_w$. Therefore $\ci w$ has the advertised Hilbert series,
since $\cj w$ does. We already knew that $I_w$ determines $\ol X_w$ as a
set; since we now know (again by Lemma \ref{lemma:IN}) that $I_w$ is
reduced, we conclude that $I_w = I(\ol X_w)$.
Having already proved that the minors form a Gr\"obner basis, so that
$J_w$ is the initial ideal, the $\ZZ^{2n}$-graded Hilbert series
calculation only requires checking that the numerator of $H(\cj
w;\xx,\yy)$ equals the double Grothendieck polynomial. This is
Corollary~\ref{cor:induction}, the base case $w = w_0$ being trivial.%
\end{proofof}
We could avoid checking the equidimensionality of $J_w$ by consulting an
oracle to find that the Grothendieck polynomials are the equivariant
$K$-classes of the matrix Schubert varieties (this result is not very far
from that in \cite{LSgrothVarDrap}). Indeed, then the Hilbert series of
$\ci w$ would immediately match that of $\cj w$. Interestingly, we could
then {\em conclude} the purity of $J_w$ from its squarefreeness, because
the geometrically isolated (i.e.\ non-embedded) components of initial
ideals of prime ideals automatically have the same dimension; see
\cite{KSprimeIdeals}. However, the analysis of $J_w$'s components is
worthwhile in any case for our combinatorial applications, such as
Theorem \ref{thm:mitosis}.
% Sections~\ref{sub:lift}--\ref{sub:alex} are characteristic free,
% the degree calculation in Section~\ref{sub:oracle} uses equivariant
% cohomology for complex varieties. Nonetheless,
% the Gr\"obner basis
% defines a flat deformation over~$\ZZ$, and the characteristic turns out
% to be inconsequential, in the end:
% Although we work over the complex numbers so as to able to relate our
% methods to equivariant cohomology, no part of our proof actually uses
% the fact that $\CC$ has characterstic zero. Nonetheless, the next
% corollary also follows from the {\em statement} of
% Theorem~\ref{thm:gb}.
%
% \begin{cor} \label{cor:char}
% Theorem~\ref{thm:gb} remains valid when $\CC$ is replaced by any other
% field.
% \end{cor}
% \begin{proof}
% Since the coefficients of the minors in $I_w$ are all integers and the
% leading coefficients are all $\pm 1$, each loop of the division
% algorithm in Buchberger's criterion \cite[Theorem~15.8]{Eis} works over
% $\ZZ$, and therefore over any ring. The only reason for restricting to
% fields is to make sense of Hilbert series.
% \end{proof}
\begin{remark}
The Gr\"obner basis in Theorem~\ref{thm:gb} defines a flat deformation
over any ring, because all of the coefficients of the minors in $I_w$ are
integers, and the leading coefficients are all $\pm 1$. Indeed, each
loop of the division algorithm in Buchberger's criterion
\cite[Theorem~15.8]{Eis} works over $\ZZ$, and therefore over any ring.
The only reason for our restricting to fields~$\kk$ is to make sense of
Hilbert series.
\end{remark}
The reader is advised to skim the rest of Section~\ref{sec:gb} upon first
reading. Although the main {\em results}\/ will be applied in later
sections, the {\em methods}\/ will find direct applications in relatively
few places. The only substantial exception is Section~\ref{sub:rc},
which translates the Stanley--Reisner theory for $J_w$ into the language
of rc-graphs via the combinatorics of antidiagonals.
\subsection{Antidiagonals and mutation}\label{sub:mutation}%%%%%%%%%%%%%%
In this subsection we begin investigating the combinatorial properties of
the monomials outside $J_w$ and the antidiagonals generating~$J_w$. For
the rest of this subsection, fix a permutation $w$ and a transposition
$s_i$ satisfying $\length(ws_i) < \length(w)$.
Define the \bem{rank matrix} $\rk(w)$ to have $(q,p)$ entry equal to
$\rank(w^T\sub qp)$. There are two standard facts we'll need concerning
rank matrices, both proved simply by looking at the picture of
$(ws_i)^T$ in~(\ref{eq:w}).
\begin{lemma} \label{lemma:rank}
Suppose the $(i,j)$ entry of\/ $\rk(ws_i)$ is $r$.
\begin{thmlist}
\item \label{i-1} If $j \geq w(i+1)$ then the $(i-1,j)$ entry of\/
$\rk(ws_i)$ is $r-1$.
\item \label{i+1}
% If $j < w(i)$, e.g.\
If $j < w(i+1)$ then the $(i+1,j)$ entry of\/ $\rk(ws_i)$ is $r$.
\end{thmlist}
\end{lemma}
In what follows, a \bem{rank condition} refers to a statement requiring
``$\rank(Z\sub qp) \leq r$'' for some $r \geq 0$. Most often, $r$ will
be either $\rank(w^T\sub qp)$ or $\rank((ws_i)^T\sub qp)$, thereby making
the entries of $\rk(w)$ and $\rk(ws_i)$ into rank conditions. We say
that a rank condition $\rank(Z\sub qp) \leq r$ \bem{causes} an
antidiagonal $a$ of the generic matrix $Z$ if $Z\sub qp$ contains $a$ and
the number of variables in $a$ is strictly larger than~$r$. For
instance, when the rank condition is in $\rk(w)$, the antidiagonals it
causes are precisely those $a \in J_w$ that are contained in $Z\sub qp$.
Although antidiagonals in $Z$ (i.e.\ antidiagonals of square submatrices
of the generic matrix $Z$) are by definition monomials, we routinely
identify each antidiagonal with its \bem{support}\/: the subset of the
\hbox{variables dividing it in $\kk[\zz]$.}
\begin{lemma} \label{lemma:i}
Antidiagonals in $J_w \minus J_{ws_i}$ are subsets of $Z\sub i{w(i)}$
and intersect row~$i$.
\end{lemma}
\begin{proof}
If an antidiagonal in $J_w$ is either contained in $Z\sub {i-1}{w(i)}$ or
not contained in $Z\sub i{w(i)}$, then some rank condition causing it is
in both $\rk(w)$ and $\rk(ws_i)$. Indeed, it is easy to check that the
rank matrices $\rk(ws_i)$ and $\rk(w)$ differ only in row $i$ between the
columns $w(i+1)$ and $w(i)-1$, inclusive.
\end{proof}
Though simple, the next lemma is the key combinatorial observation. Note
that the permutation $w$ there is arbitrary; in particular, we will
frequently apply the lemma in the context of antidiagonals for a
permutation called $ws_i$.
\begin{lemma} \label{lemma:squish}
Suppose $a \in J_w$ is an antidiagonal and $a' \subset Z$ is another
antidiagonal.
\begin{enumerate}
\item[{\makebox[3.5ex][c]{(W)}}] \label{west} If $a'$ is obtained by
moving west one or more of the variables in $\,a$, then $a' \in J_w$.
\item[{\makebox[3.5ex][c]{(E)}}] \label{east} If $a' \in \kk[\zz]$ is
obtained by moving east any variable {\em except} the northeast one in
$\,a$, then $a' \in J_w$.
\item[{\makebox[3.5ex][c]{(N)}}] \label{north} If $a'$ is obtained by
moving north one or more of the variables in $\,a$, then $a' \in J_w$.
\item[{\makebox[3.5ex][c]{(S)}}] \label{south} If $a' \in \kk[\zz]$ is
obtained by moving south any variable {\em except} the southwest one~in
$\,a$, then $a' \in J_w$.
\end{enumerate}
\end{lemma}
\begin{proof}
Every rank condition causing $a$ also causes any of the antidiagonals
$a'$.
\end{proof}
\begin{example} \label{ex:squish}
Parts~(W) and~(E) of Lemma~\ref{lemma:squish} together imply that the
type of motion depicted in the following diagram preserves the property
of an antidiagonal being in~$J_w$. The presence of the northeast $*$
justifies moving the southwest $*$ east.
$$
\begin{array}{@{}r@{\;}c@{\;}l@{}}
&&\adots\\&
% 1 2 3 4 5 6
\begin{array}{@{}l@{}|c|c|@{}c@{}|c|c|@{}}
\cline{2-6}
& & & \toplinedots & \phantom{*} & *
\\\cline{2-3}\cline{5-6}
&*& \phantom{*} & & &
\\\cline{2-6}
\end{array}\\
\adots
\end{array}
\in J_w \quad \implies \quad
\begin{array}{@{}r@{\;}c@{\;}l@{}}
&&\adots\\&
% 1 2 3 4 5 6
\begin{array}{@{}l@{}|c|c|@{}c@{}|c|c|@{}}
\cline{2-6}
& & & \toplinedots & \phantom{*}
& \makebox[0mm][r]{$\leftarrow$\hspace{1pt}}*
\\\cline{2-3}\cline{5-6}
&*\makebox[0mm][l]{\hspace{1pt}$\rightarrow$}
&\phantom{*} & & &
\\\cline{2-6}
\end{array}\\
\adots
\end{array}
\in J_w
$$
The two rows could also be separated by some other rows---possibly
themselves containing elements of the original antidiagonal---as long as
the indicated motion preserves the fact that we have an antidiagonal.
\end{example}
\begin{defn} \label{defn:mu}
Let $\bb$ be an array $\bb = (b_{rs})$ of nonnegative integers, i.e.\
$\bb$ is an \bem{exponent array} for a monomial $\zz^\bb \in \kk[\zz]$.
Let
\begin{eqnarray*}
\west_q(\bb) &:=& \min(\{p \mid b_{qp} \neq 0\}\cup\{\infty\})
\end{eqnarray*}
be the column of the leftmost (most ``western'') nonzero entry in row
$q$. Define the \bem{mutation} of $\bb$,
\begin{eqnarray*}
\mu_i(\bb) &:=& \hbox{\rm the exponent array of }
({z_{i,p}}/{z_{i+1,p}}) \zz^\bb \hbox{\rm\ for } p = \west_{i+1}(\bb).
\end{eqnarray*}
For ease of notation, we write $\mu_i(\zz^\bb)$ for
$\zz^{\mu_i(\bb)}$.
\end{defn}
If one thinks of the array $\bb$ as a chessboard with some coins stacked
in each square, then $\mu_i$ is performed by taking a coin off the
western stack in row $i+1$ and putting it onto the stack due north of it
in row $i$.
\begin{example} \label{ex:mutate}
Suppose $\bb$ is the left array below and $i = 3$. In Fig.~\ref{fig:mu}
we list (reading left to right as usual) 7 mutations of $\bb$, namely
$\bb = (\mu_3)^0(\bb)$ through $(\mu_3)^6(\bb)$ (after that it involves
the dots we left unspecified). Here, the empty boxes denote entries
equal to $0$, and the nonzero mutated entries at each step are in
boldface. To make things easier to look at, the entries on or below the
main antidiagonal are represented by dots, each of which may be zero or
not (independently of the others). The $3$ and~$4$ at left are labels
for rows $i = 3$ and $i+1 = 4$.
\begin{figure}[t]
\begin{rcgraph}
\begin{array}{@{}cc@{}}{}\\\\3&\\4\\\\\\\\\\\end{array}
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline 1 & & 1 & &\ \, & 1 & 1 &\hspace{1pt}\cdot\hspace{1pt}
\\\hline 1 & & 1 & 1 & 1 &\ \, &\cdot&\cdot
\\\hline & &\ \, & & 1 &\cdot&\cdot&\cdot
\\\hline 2 & 2 &\ \, & 2 &\cdot&\cdot&\cdot&\cdot
\\\hline & 1 & 1 &\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline & &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline
\end{array}
\ \stackrel{\textstyle \mu_3}{\longmapsto}\,
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline 1 & & 1 & &\ \, & 1 & 1 &\hspace{1pt}\cdot\hspace{1pt}
\\\hline 1 & & 1 & 1 & 1 &\ \, &\cdot&\cdot
\\\hline\bf 1& &\ \, & & 1 &\cdot&\cdot&\cdot
\\\hline\bf 1& 2 &\ \, & 2 &\cdot&\cdot&\cdot&\cdot
\\\hline & 1 & 1 &\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline & &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline
\end{array}
\ \stackrel{\textstyle \mu_3}{\longmapsto}\,
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline 1 & & 1 & &\ \, & 1 & 1 &\hspace{1pt}\cdot\hspace{1pt}
\\\hline 1 & & 1 & 1 & 1 &\ \, &\cdot&\cdot
\\\hline\bf 2& &\ \, & & 1 &\cdot&\cdot&\cdot
\\\hline & 2 &\ \, & 2 &\cdot&\cdot&\cdot&\cdot
\\\hline & 1 & 1 &\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline & &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline
\end{array}
\end{rcgraph}
\begin{rcgraph}
\phantom{
\begin{array}{@{}cc@{}}{}\\\\3&\\4\\\\\\\\\\\end{array}
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline 1 & & 1 & &\ \, & 1 & 1 &\hspace{1pt}\cdot\hspace{1pt}
\\\hline 1 & & 1 & 1 & 1 &\ \, &\cdot&\cdot
\\\hline & &\ \, & & 1 &\cdot&\cdot&\cdot
\\\hline 2 & 2 &\ \, & 2 &\cdot&\cdot&\cdot&\cdot
\\\hline & 1 & 1 &\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline & &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline
\end{array}}
\ \stackrel{\textstyle \mu_3}{\longmapsto}\,
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline 1 & & 1 & &\ \, & 1 & 1 &\hspace{1pt}\cdot\hspace{1pt}
\\\hline 1 & & 1 & 1 & 1 &\ \, &\cdot&\cdot
\\\hline 2 &\bf 1&\ \, & & 1 &\cdot&\cdot&\cdot
\\\hline &\bf 1&\ \, & 2 &\cdot&\cdot&\cdot&\cdot
\\\hline & 1 & 1 &\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline & &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline
\end{array}
\ \stackrel{\textstyle \mu_3}{\longmapsto}\,
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline 1 & & 1 & &\ \, & 1 & 1 &\hspace{1pt}\cdot\hspace{1pt}
\\\hline 1 & & 1 & 1 & 1 &\ \, &\cdot&\cdot
\\\hline 2 &\bf 2&\ \, & & 1 &\cdot&\cdot&\cdot
\\\hline & &\ \, & 2 &\cdot&\cdot&\cdot&\cdot
\\\hline & 1 & 1 &\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline & &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline
\end{array}
\end{rcgraph}
\begin{rcgraph}
\phantom{
\begin{array}{@{}cc@{}}{}\\\\3&\\4\\\\\\\\\\\end{array}
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline 1 & & 1 & &\ \, & 1 & 1 &\hspace{1pt}\cdot\hspace{1pt}
\\\hline 1 & & 1 & 1 & 1 &\ \, &\cdot&\cdot
\\\hline & &\ \, & & 1 &\cdot&\cdot&\cdot
\\\hline 2 & 2 &\ \, & 2 &\cdot&\cdot&\cdot&\cdot
\\\hline & 1 & 1 &\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline & &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline
\end{array}}
\ \stackrel{\textstyle \mu_3}{\longmapsto}\,
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline 1 & & 1 & &\ \, & 1 & 1 &\hspace{1pt}\cdot\hspace{1pt}
\\\hline 1 & & 1 & 1 & 1 &\ \, &\cdot&\cdot
\\\hline 2 & 2 &\ \, &\bf 1& 1 &\cdot&\cdot&\cdot
\\\hline & &\ \, &\bf 1&\cdot&\cdot&\cdot&\cdot
\\\hline & 1 & 1 &\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline & &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline
\end{array}
\ \stackrel{\textstyle \mu_3}{\longmapsto}\,
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline 1 & & 1 & &\ \, & 1 & 1 &\hspace{1pt}\cdot\hspace{1pt}
\\\hline 1 & & 1 & 1 & 1 &\ \, &\cdot&\cdot
\\\hline 2 & 2 &\ \, &\bf 2& 1 &\cdot&\cdot&\cdot
\\\hline & &\ \, & &\cdot&\cdot&\cdot&\cdot
\\\hline & 1 & 1 &\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline & &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline &\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\\\hline
\end{array}
\end{rcgraph}
\caption{Mutation} \label{fig:mu}
\end{figure}
\end{example}
Having chosen our permutation $w$ and row index $i$, various entries of a
given $\bb \in \ZZ^{n^2}$ play special roles. To begin with, we call the
union of rows $i$ and $i+1$ the \bem{gene}%
%
\footnote{All of the unusual terminology in what follows comes
from genetics. Superficically, our diagrams with two rows of
boxes look like geneticists' schematic diagrams of the DNA double
helix; but there is a much more apt analogy that will become
clear only in Section~\ref{sub:coarsen}, where the biological
meanings of the terms can be found in another footnote.}
%
of $\bb$. For exponent arrays $\bb$ such that $\zz^\bb \not\in J_w$, the
spot in row $i$ and column
\begin{eqnarray} \label{eq:start}
\start_i(\bb) &:=& \min\{p \mid z_{ip}\zz^\bb \not\in J_w\}
\end{eqnarray}
is called the \bem{start codon} of $\bb$. The minimum defining
$\start_i(\bb)$ is taken over a nonempty set because $J_w \subseteq J_{w_0}
= I_{w_0} = \$, so that $z_{in}\zz^\bb$ remains
outside of $J_w$. Of course, $z_{ip}\zz^\bb \not\in J_w$ whenever
$z_{ip}$ divides $\zz^\bb \not\in J_w$ because $J_w$ is generated by
squarefree monomials. Therefore,
\begin{eqnarray} \label{eq:leq}
\start_i(\bb) &\leq& \west_i(\bb).
\end{eqnarray}
For completeness, set $\start_i(\bb)=0$ if $\zz^\bb \in J_w$.
Also of special importance is the \bem{promoter} $\prom(\bb)$, consisting
of the rectangular $2 \times (\start_i(\bb)-1)$ array of locations in the
gene of $\bb$ that are strictly west of $\start_i(\bb)$. Again, we omit
the explicit reference to $i$ and $w$ in the notation because these are
fixed for the discussion. The sum of all entries in the promoter of
$\bb$ is
\begin{eqnarray} \label{eq:loose}
|\prom(\bb)| &=& \sum_{j < \start_i(\bb)} b_{i+1,j}.
\end{eqnarray}
% operator $\ddem iw$ in the following definition is the identity on a
% monomial $\zz^\bb$ when row $i$ of $\bb$ extends west of row $i+1$.
\begin{example} \label{ex:prom}
Let $\bb$ be the left array in Example~\ref{ex:mutate}, $i = 3$, and $w =
13865742$, the permutation displayed in Example~\ref{ex:intro}. Then the
gene of $\bb$ consists of rows $i = 3$ and $i+1 = 4$, and we claim
$\start_i(\bb) = 5$.
To begin with, we have $6$ choices for an antidiagonal $a \in J_w$
dividing $z_{31}\zz^\bb$: we must have $z_{31} \in a$, but other than
that we are free to choose one element of $\{z_{23}, z_{24}, z_{25}\}$
and one element of $\{z_{16},z_{17}\}$. (This gives an example of the
$a$ produced in the first paragraph of the proof of
Lemma~\ref{lemma:outside}, below.) Even more varied choices are
available for $z_{32}\zz^\bb$, such as $z_{41}z_{32}z_{23}$ or
$z_{41}z_{32}z_{13}$. We can similarly find lots of antidiagonals in
$J_w$ dividing $z_{33}\zz^\bb$, and $z_{34}\zz^\bb$. On the other hand,
$z_{35}$ already divides $\zz^\bb$, and one can verify that $\zz^\bb$
isn't divisible by the antidiagonals of any of the $2 \times 2$ or $3
\times 3$ minors defining $I_w$ (see Example~\ref{ex:intro}). Therefore
$z_{35}\zz^\bb \not\in J_w$, so $\start_i(\bb) = 5$.
The promoter $\prom(\bb)$ consists of the $2 \times 4$ block
\begin{rcgraph}
\begin{array}{l|c|c|c|c|}
\cline{2-5} 3\ & & & &
\\\cline{2-5} 4\ & 2 & 2 &\ \, & 2
\\\cline{2-5}
\end{array}
\end{rcgraph}
at the western end. In particular, $|\prom(\bb)| = 6$.
Nothing in this example depends on the values chosen for the dots on or
below the main antidiagonal.%
\end{example}
\subsection{Lifting Demazure operators}\label{sub:lift}%%%%%%%%%%%%%%%%%%
Now we need to understand the Hilbert series of $\cj w$ for varying $w$.
Since $J_w$ is a monomial ideal, it makes sense to consider its
$\ZZ^{n^2}$-graded Hilbert series $H(\cj w; \zz)$, which we express in
the variables $(z_{ij})$. Observe that $H(\cj w; \zz)$ is simply the sum
of all monomials outside of $J_w$. Using the combinatorics of the
previous subsection, we construct operators $\ddem iw$ defined on
monomials and taking the power series $H(\cj w; \zz)$ to $H(\cj{ws_i};
\zz)$ whenever $\length(ws_i) < \length(w)$. In other words, the sum of
all monomial outside $J_{ws_i}$ is obtained from the sum of monomials
outside $J_w$ by replacing $\zz^\bb \not\in J_w$ with $\ddem
iw(\zz^\bb)$. It is worth keeping in mind that we will eventually show
(in Section~\ref{sub:coarsen}) how $\ddem iw$ refines the usual
$\ZZ^n$-graded Demazure operator $\dem i$, when these operators are
applied to the variously graded Hilbert series of $\cj w$.
Again, fix for the duration of this subsection a permutation $w$ and a
transposition $s_i$ satisfying $\length(ws_i) < \length(w)$ .
\begin{defn} \label{defn:ddem}
The \bem{lifted Demazure operator} corresponding to $w$ and $i$ is a map
of abelian groups $\ddem iw : \ZZ[[\zz]] \too \ZZ[[\zz]]$ determined by
its action on monomials:
\begin{eqnarray*}
\ddem iw(\zz^\bb) &:=& \sum_{d=0}^{|\prom(\bb)|} {\mu_i}^d(\zz^\bb).
\end{eqnarray*}
Here, ${\mu_i}^d$ means take the result of applying $\mu_i$ a total of $d$
times, and ${\mu_i}^0(\bb) = \bb$.
\end{defn}
\begin{example} \label{ex:ddem}
If $\bb$ is the array in Examples~\ref{ex:mutate} and~\ref{ex:prom}, then
$\ddem 3w(\zz^\bb)$ is the sum of the $7$ monomials whose exponent arrays
are displayed in Example~\ref{ex:mutate}.%
% This statement is independent of the values chosen for the dots on or
% below the main antidiagonal.%
\end{example}
Observe that $\ddem iw$ replaces each monomial by a homogeneous
polynomial of the same total degree, so the result of applying $\ddem iw$
to a power series is actually a power~series.
In preparation for Proposition~\ref{prop:ev}, we need a few lemmas
detailing the effects of mutation on monomials and their genes. The
first of these implies that $\ddem iw$ takes monomials outside $J_w$ to
sums of monomials outside $J_{ws_i}$, given that ${\mu_i}^0(\bb) = \bb$.
\begin{lemma} \label{lemma:outside}
If $\zz^\bb \not\in J_w$ and $1 \leq d \leq |\prom(\bb)|$ then
${\mu_i}^d(\zz^\bb) \in J_w \minus J_{ws_i}$.
\end{lemma}
\begin{proof}
We may as well assume $|\prom(\bb)| \geq 1$, or else the statement is
vacuous. By definition of $\prom(\bb)$ and $\start_i(\bb)$, some
antidiagonal $a \in J_w$ divides $z_{ip}\zz^\bb$, where here (and for the
remainder of this proof) $p = \west_{i+1}(\bb)$. Since $a$ doesn't
divide $\zz^\bb$, we find that $z_{ip} \in a$, whence $a$ cannot
intersect row $i+1$, which is zero to the west of~$z_{ip}$. Thus $a$
also divides $\mu_i(\zz^\bb)$, and hence ${\mu_i}^d(\zz^\bb)$ for all $d$
(including $d > |\prom(\bb)|$, but we won't need this).
It remains to show that ${\mu_i}^d(\zz^\bb) \not\in J_{ws_i}$ when $d
\leq |\prom(\bb)|$. Let's start with $d \leq b_{i+1,p}$. Any
antidiagonal $a$ dividing ${\mu_i}^d(\zz^\bb)$ does not continue
southwest of~$z_{ip}$; this is by Lemma~\ref{lemma:squish}(S)
and the fact that $\zz^\bb \not\in J_w$ (we could move $z_{ip}$ south).
Suppose for contradiction that $a \in J_{ws_i}$, and consider the
smallest northwest submatrix $Z\sub i{j(a)}$ containing $a$. If $j(a)
\geq w(i+1)$ then the antidiagonal $a' = a/z_{ip}$ obtained by omitting
$z_{ip}$ from $a$ is still in $J_{ws_i}$, being caused by the entry of
$\rk(ws_i)$ at $(i-1,j(a))$ as per Lemma~\ref{lemma:rank}.\ref{i-1}. On
the other hand, if $j(a) < w(i+1)$, then $a'' = (z_{i+1,p}/z_{ip})a$ is
still in $J_{ws_i}$, being caused by the entry of $\rk(ws_i)$ at
$(i+1,j(a))$ as per Lemma~\ref{lemma:rank}.\ref{i+1}. Since both $a'$
and $a''$ divide $\zz^\bb$ by construction, we find that $\zz^\bb \in
J_{ws_i} \subset J_w$, the desired contradiction. It follows that
${\mu_i}^d(\zz^\bb) \not\in J_{ws_i}$ for $d \leq b_{i+1,p}$.
Assuming the result for $d \leq \sum_{j=p}^{p'} b_{i+1,j}$, where $p' <
\start_i(\bb) - 1$, we now demonstrate the result for $d \leq
\sum_{j=p}^{p'+1} b_{i+1,j}$. Again, any antidiagonal $a \in J_w$
dividing ${\mu_i}^d(\zz^\bb)$ must end at row $i$, for the same reason as
in the previous paragraph. But now if $a \in J_{ws_i}$, then moving its
southwest variable to $z_{ip}$ creates an antidiagonal that is in
$J_{ws_i}$ (by Lemma~\ref{lemma:squish}(W)) and divides
$\mu_i(\zz^\bb)$, which we have seen is impossible.%
\end{proof}
Now we show that mutation of monomials outside $J_w$ can't produce the
same monomial more than once, as long we stop after $|\prom|$ many steps.
\begin{lemma} \label{lemma:unique}
Suppose $\zz^\bb, \zz^{\bb'} \not\in J_w$ and that $d,d' \in \ZZ$ satisfy
$1 \leq d \leq |\prom(\bb)|$ and $1 \leq d' \leq |\prom(\bb')|$. If $\bb
\neq \bb'$ then ${\mu_i}^d(\bb) \neq {\mu_i}^{d'}(\bb')$.
\end{lemma}
\begin{proof}
The inequality $d \leq |\prom(\bb)|$ guarantees that the mutations of
$\bb$ only alter the promoter of $\bb$, which is west of $\west_i(\bb)$
by~(\ref{eq:leq}). Therefore, assuming (by switching $\bb$ and $\bb'$ if
necessary) that $\west_i(\bb') \leq \west_i(\bb)$, we reduce to the case
where $\bb$ and $\bb'$ differ only in their genes, in columns strictly
west of $\west_i(\bb)$.
Let $\cc = {\mu_i}^d(\bb)$ and $\cc' = {\mu_i}^{d'}(\bb')$. Mutating
preserves the sums
$$
b_{i+1,j} = c_{ij} + c_{i+1,j} \quad {\rm and} \quad c'_{ij} +
c'_{i+1,j} = b_{ij} + b_{i+1,j}
$$
for $j < \west_i(\bb)$, and we may as well assume these are equal for
every $j$, or else $\cc \neq \cc'$ is clear. The westernmost column
where $\bb$ and $\bb'$ disagree is now necessarily $p = \west_i(\bb')$.
It follows that $z_{ip}\zz^\bb \not\in J_w$, because $\bb$ agrees with
$\bb'$ strictly to the west of column~$p$ as well as strictly to the
north of row $i$, and any antidiagonal $a \in J_w$ dividing
$z_{ip}\zz^\bb$ must be contained in this region (since it contains
$z_{ip}$). In particular, $\start_i(\bb) \leq p$. We conclude that
mutating $\bb$ and $\bb'$ fewer than $|\prom(\bb)|$ or $|\prom(\bb')|$
times cannot alter the column $p$ where $\bb$ and $\bb'$ differ. Thus
$\cc$ differs from $\cc'$ in column$~p$.
\end{proof}
\begin{example}
If we apply $\mu_i$ more than $|\prom(\bb)|$ times to some array $\bb$,
it is possible to reach ${\mu_i}^{d'}(\bb')$ for some $\bb' \neq \bb$ and
$d' \leq |\prom(\bb')|$. Take $\bb$, $i$, and $w$ as in
Examples~\ref{ex:mutate}, \ref{ex:prom}, and~\ref{ex:ddem}, and set the
dot in $\bb$ at position $(4,5)$ equal to $3$. If $\zz^{\bb'} =
(z_{35}/z_{45})\zz^\bb$, then we have $|\prom(\bb)| = |\prom(\bb')| = 6$,
but the entries of $\bb$ and $\bb'$ in column $5$ of their genes are $1
\atop 3$ and $2 \atop 2$, respectively. Mutating $\bb$ and $\bb'$ up to
$6$ times yields $7$ arrays each, all distinct because of the $1 \atop 3$
and $2 \atop 2$ in column $5$. However, mutating $\bb$ to an $8^\th$
array $(\mu_3)^7(\bb)$ changes the $1 \atop 3$ to $2 \atop 2$, and
outputs $(\mu_3)^6(\bb')$.
\end{example}
If $\cc$ is the result of applying $\mu_i$ to $\bb$ some number of times,
we can recover $\bb$ from $\cc$ by \bem{reverting} certain entries of
$\cc$ from row $i$ back to row $i+1$. Formally, reverting an entry
$c_{ij}$ of $\cc$ means making a new array that agrees with $\cc$ except
at $(i,j)$ and $(i+1,j)$. In those spots, the new array has $(i,j)$
entry $0$ and $(i+1,j)$ entry $c_{ij} + c_{i+1,j}$. (In terms of the
stacks-of-coins picture, we revert only entire stacks of coins, not
single coins.) Even if we're just given $\cc$ without knowing $\bb$, we
still have a criterion to determine when a certain reversion of $\cc$
yields a monomial $\zz^\bb \not\in J_w$.
\begin{claim} \label{claim:min}
Suppose $\zz^\cc \in J_w \minus J_{ws_i}$. If\/ $\bb$ is obtained from
$\cc$ by reverting all entries of $\cc$ in row $i$ that are west of or at
column $\west_{i+1}(\cc)$, then $\zz^\bb \not\in J_w$.
% In particular, $\west_i(\cc) \leq \west_{i+1}(\cc)$.
\end{claim}
\begin{proof}
Suppose $\zz^\bb \in J_w$, and let's try to produce an antidiagonal
witness $a \in J_w$ dividing it. Either $a$ ends at row $i$, or not. In
the first case, $a$ divides $\zz^\cc$, because the nonzero entries in row
$i$ of $\bb$ are the same as the corresponding entries of $\cc$. Thus we
can replace $a$ by the result $a'$ of tacking on $z_{i+1,p}$ to $a$,
where $p = \west_{i+1}(\cc)$. This new $a'$ is in $J_w$
% by Lemma~\ref{lemma:rank}.\ref{leq} (applied to $\rk(w)$, not
% $\rk(ws_i)$),
because $a \in J_w$ divides $a'$. It follows from Lemma~\ref{lemma:i}
that $a' \in J_{ws_i}$. Furthermore, $a'$ divides $\zz^\cc$ by
construction, and thus contradicts our assumption that $\zz^\cc \not\in
J_{ws_i}$. Therefore, we may assume for the remainder of the proof of
this lemma that $a$ doesn't end at row $i$, so $a \in J_{ws_i}$ by
Lemma~\ref{lemma:i}.
We now prove that $\zz^{\bb} \not\in J_w$ by showing that if $a \in
J_{ws_i}$ and $a$ divides $\zz^{\bb}$, then from $a$ we can synthesize
$a' \in J_{ws_i}$ dividing $\zz^\cc \in J_{ws_i}$, again contradicting
our running assumption $\zz^\cc \in J_w \minus J_{ws_i}$. There are
three possibilities (an illustration for~(\ref{ii}) is described in
Example~\ref{ex:ii}):
\begin{textlist}
\item \label{i} The antidiagonal $a \in J_{ws_i}$ intersects row $i$ but
doesn't end there.
\item \label{ii} The antidiagonal $a \in J_{ws_i}$ skips row $i$ but
intersects row $i+1$.
\item \label{iii} The antidiagonal $a \in J_{ws_i}$ skips both row $i$ as
well as row $i+1$.
\end{textlist}
In case~(\ref{i}) either $a$ already divides $\zz^\cc$ or we can move
east the row $i+1$ variable in $a$, into the location
$(i+1,\west_{i+1}(\cc))$. The resulting antidiagonal $a'$ divides $\cc$
by construction and is in $J_{ws_i}$ by
Lemma~\ref{lemma:squish}(E). In case~(\ref{iii}) the
antidiagonal already divides $\zz^\cc$ because $\bb$ agrees with $\cc$
outside of their genes.
This leaves case~(\ref{ii}). If $a$ doesn't already divide $\zz^\cc$,
then the intersection $z_{i+1,j}$ of $a$ with row $i+1$ is strictly west
of $\west_{i+1}(\cc)$. The antidiagonal $a' = (z_{ij}/z_{i+1,j})a$ then
divides $\zz^\cc$ by construction, and is in $J_{ws_i}$ by
Lemma~\ref{lemma:squish}(N).
\end{proof}
\begin{example} \label{ex:ii}
Here is an example of what happens in case~(\ref{ii}) from the proof of
Claim~\ref{claim:min}. Let $\bb$ and $\cc$ be the first and last arrays
from Example~\ref{ex:mutate}, and consider what happens when we fiddle
with their $(5,1)$ entries. The antidiagonals $z_{51}z_{42}z_{23}$ and
$z_{51}z_{42}z_{24} \in J_{ws_i}$ both divide $z_{51}\zz^\bb$. Using
Lemma~\ref{lemma:squish}(N) we can move the $z_{42}$ north to
$z_{32}$ to get $a' \in \{z_{51}z_{32}z_{23}, z_{51}z_{32}z_{24}\}$ in
$J_{ws_i}$ dividing $z_{51}\zz^\cc$. (It almost goes without saying, of
course, that $z_{51}\zz^\cc$ is no longer in $J_w \minus J_{ws_i}$, so
it doesn't satisfy the hypothesis of Claim~\ref{claim:min}; we were,
after all, looking for a contradiction.)
\end{example}
Any array $\cc$ whose row $i$ begins west of its row $i+1$ can be
expressed as a mutation of {\em some} array $\bb$. By
Claim~\ref{claim:min}, we even know how to make sure $\zz^\bb \not\in
J_w$ whenever $\zz^\cc \in J_w \minus J_{ws_i}$. But we also want each
$\zz^\cc \in J_w \minus J_{ws_i}$ to appear in $\ddem iw(\zz^\bb)$ for
some $\zz^\bb \not\in J_w$, and this involves making sure $\start_i(\bb)$
isn't too far west.
\begin{example} \label{ex:revert}
If $\west_i(\cc)$ is sufficiently smaller than $\west_{i+1}(\cc)$, then
it might be hard to determine which entries in row $i$ of $\cc$ to revert
while still assuring that $\zz^\cc$ appears in $\ddem iw(\zz^\bb)$. For
example, let $\cc$ be the last array in Example~\ref{ex:mutate}, that is,
$\cc = (\mu_3)^6(\bb)$. Suppose further that the dot at $(4,5)$,
%---that is, southeast of the boldface ${\b f 2}$---
% or all of the dots in row $4$, are
is really blank---i.e.\ zero. Without a priori knowing $\bb$, how are we
to know {\em not} to revert the $1$ in position $(3,5)$? Well, let's
suppose we did revert this entry, along with all of the entries west of
it in row $3$. Then we'd end up with an array $\bb'$ such that
$\zz^{\bb'} \not\in J_w$ all right, as per Claim~\ref{claim:min}, but
also such that $z_{35}\zz^{\bb'} \not\in J_w$. This latter condition is
intolerable, since $5 \geq \start_i(\bb)$ implies that our original
$\zz^\cc$ won't end up in the sum $\ddem 3w(\zz^{\bb'})$.
Thus the problem with trying to revert the $1$ in position $(3,5)$ is
that it's too far east. On the other hand, we might also try reverting
only the row $3$ entries in columns $1$ and $2$, but with dire
consequences: we end up with an array $\bb''$ such that $\zz^{\bb''}$ is
divisible by $z_{42}z_{34}z_{25} \in J_w$. (This is an example of the
antidiagonal $a$ to be produced after the displayed equation in the proof
of Lemma~\ref{lemma:revert}.) We are left with only one choice: revert
the boldface ${\mathbf 2}$ in position $(3,4)$ and all of its more
western brethren.%
\end{example}
In general, as in the previous example, the \bem{critical column} for
$\zz^\cc \in J_w \minus J_{ws_i}$ is
\begin{eqnarray*}
\crit(\cc) &:=& \min(p \leq \west_{i+1}(\cc)\mid z_{ip}\hbox{ divides }
\zz^\cc \hbox{ and } z_{i+1,p}\zz^\cc \not\in J_{ws_i}).
\end{eqnarray*}
\begin{claim} \label{claim:crit}
If $\zz^\cc \in J_w \minus J_{ws_i}$ then:
\begin{thmlist}
\item \label{crit1}
the set used to define $\crit(\cc)$ is nonempty;
\item \label{crit2}
reverting $c_{i,\crit(\cc)}$ creates an array $\cc'$ such that
$\zz^{\cc'} \not\in J_{ws_i}$; and
\item \label{crit3}
if $\west_i(\cc) < \crit(\cc)$, then the monomial $\zz^{\cc'}$ from
part~\ref{crit2} remains in $J_w$.
\end{thmlist}
\end{claim}
\begin{proof}
Claim~\ref{claim:min} implies $\west_i(\cc) \leq \west_{i+1}(\cc)$, so
$p' = \max(p \leq \west_{i+1}(\cc) \mid c_{ip} \neq 0)$ is well-defined.
If $a$ is an antidiagonal dividing the monomial whose exponent array is
the result of reverting $c_{ip'}$, then $a$ divides either $\zz^\cc$ or
the monomial $\zz^\bb$ from Claim~\ref{claim:min}, and neither of these
is in $J_{ws_i}$. Thus $a \not\in J_{ws_i}$ and part~\ref{crit1} is
proved. Part~\ref{crit2} is by definition, and part~\ref{crit3} follows
from it by Lemmas~\ref{lemma:i} and~\ref{lemma:squish}(W).
\end{proof}
\begin{lemma} \label{lemma:revert}
Suppose $\zz^\cc \in J_w \minus J_{ws_i}$ and that $\bb$ is obtained
by reverting all row $i$ entries of $\cc$ west of or at $\crit(\cc)$.
Then $\zz^\bb \not\in J_w$, and $\crit(\cc) < \start_i(\bb)$.
\end{lemma}
\begin{proof}
The proof that this $\zz^\bb$ is not in $J_w$ has two cases. In the
first case we have $\crit(\cc) = \west_{i+1}(\cc)$, and
Claim~\ref{claim:min} immediately implies the result. In the second case
we have $\crit(\cc) < \west_{i+1}(\cc)$, and we can apply
Claim~\ref{claim:min} to the monomial $\zz^{\cc'} \in J_w \minus
J_{ws_i}$ from Claim~\ref{claim:crit}.
Now we need to show $z_{ip}\zz^\bb \in J_w$ for two kinds of $p$: for $p
\leq \west_i(\cc)$ and $\west_i(\cc) < p \leq \crit(\cc)$. (Of course,
when $\west_i(\cc) = \crit(\cc)$ the second of these cases is vacuous.)
The case $p \leq \west_i(\cc)$ is a little easier, so we treat it first.
There is some antidiagonal in $J_w$ ending on row $i$ and dividing
$\zz^\cc$, by Lemma~\ref{lemma:i}. When $p \leq \west_i(\cc)$, we get
the desired result by appealing to Lemma~\ref{lemma:squish}(W).
Next we treat $\west_i(\cc) < p \leq \crit(\cc)$. These inequalities
mean precisely that
\begin{eqnarray*}
j &=& \max\{p' < p \mid c_{ip'} \neq 0\}
\end{eqnarray*}
is well-defined, and that $z_{i+1,j}\zz^\cc \in J_{ws_i}$. Any
antidiagonal $a \in J_{ws_i}$ dividing $z_{i+1,j}\zz^\cc$ must contain
$z_{i+1,j}$ because $a$ doesn't divide $\zz^\cc$, and the fact that
$\zz^\bb \not\in J_w$ implies that $a$ also doesn't divide $\zz^\bb$. It
follows that $a$ intersects row $i$ at some spot in which $\cc$ is
nonzero strictly east of column $j$. This spot is necessarily east of or
at $(i,p)$ by construction. Without changing whether $a \in J_{ws_i}$,
Lemma~\ref{lemma:squish}(W) says that we may assume $a$ contains
$z_{ip}$ itself. This $a$ divides $z_{ip}\zz^\bb$, whence $z_{ip}\zz^\bb
\in J_{ws_i}$.
\end{proof}
The next proposition is the main result of Section~\ref{sub:lift},
pinpointing, at the level of individual standard monomials, the relation
between $J_w$ and $J_{ws_i}$.
\begin{prop} \label{prop:ev}
$_{\!}H(\cj{ws_i}; \zz) = \ddem iw H(\cj w; \zz)$ if $\length(ws_i) <
\length(w)$.
\end{prop}
\begin{proof}
We need the sum $H(\cj{ws_i}; \zz)$ of monomials outside $J_{ws_i}$ to be
obtained from the sum of monomials outside $J_w$ by replacing $\zz^\bb
\not\in J_w$ with $\ddem iw(\zz^\bb)$. We know by
Lemma~\ref{lemma:outside} that $\ddem iw H(\cj w)$ is a sum of monomials
outside $J_{ws_i}$. Furthermore, no monomial $\zz^\cc$ is repeated in
this sum: if $\zz^\cc \not\in J_w$ appears in $\ddem iw(\bb)$, then $\bb$
must equal $\cc = {\mu_i}^0(\bb)$ by Lemma~\ref{lemma:outside}; and if
$\zz^\cc \in J_w$ then Lemma~\ref{lemma:unique} applies.
It remains to demonstrate that each monomial $\zz^\cc \not\in J_{ws_i}$
is equal to ${\mu_i}^d(\zz^\bb)$ for some monomial $\zz^\bb \not\in J_w$
and $d \leq |\prom(\bb)|$. This is easy if $\zz^\cc$ isn't even in
$J_w$: we take $\zz^\bb = {\mu_i}^0(\zz^\bb) = \zz^\cc$. Since we can
now assume $\zz^\cc \in J_w \minus J_{ws_i}$, the result follows from
Lemma~\ref{lemma:revert}, once we notice that the inequality $\crit(\cc) <
\start_i(\bb)$ there is equivalent to the inequality $d \leq |\prom(\bb)|$.
\end{proof}
\subsection{Coarsening the grading}\label{sub:coarsen}%%%%%%%%%%%%%%%%%%%
As in Section~\ref{sub:lift}, fix a permutation $w$ and an index $i$ such
that $\length(ws_i) < \length(w)$. Our goal in this subsection is to
prove (in Corollary~\ref{cor:induction}) that the set of $\ZZ^n$-graded
Hilbert series $H(\cj w; \xx)$ for varying $w$ is closed under Demazure
operators. The idea is to combine lifted Demazure operators $\ddem iw$
with the \bem{specialization} map $\XX : \ZZ[[\zz]] \to \ZZ[[\xx]]$ that
sends $z_{qp} \mapsto x_q$, which we also call \bem{coarsening the
grading} from $\ZZ^{n^2}$ to $\ZZ^n$. As usual, we present the proof in
the ``single'' case, for ease of notation, but indicate which changes of
notation make the arguments work for the $\ZZ^{2n}$-graded Hilbert series
$H(\cj w;\xx,\yy)$, with the specialization $\XX_\YY : \ZZ[[\zz]] \to
\ZZ[[\xx,\yy^{-\1}]]$ sending $z_{qp} \mapsto x_q/y_p$.
At the outset, we could hope that $\XX \circ \ddem iw = \dem i \circ
\XX$ monomial by monomial. However, although this works in some cases
(see (\ref{eq:fixed}), below) it fails in general. The next lemma will
be used to take care of the general case. Its proof is somewhat involved
and irrelevant to its application, so we postpone the proof until after
Proposition~\ref{prop:lift}. (In fact, there's really no reason for
anyone to go through the proof of the lemma on their first reading.)
Denote by $\std(J_w)$ the set of \bem{standard exponent arrays\/}: the
exponents on monomials not in $J_w$.
\begin{lemma} \label{lemma:involution}
There is an involution $\tau : \std(J_w) \to \std(J_w)$ such that $\tau^2
= 1$ and:
\begin{thmlist}
\item
$\tau\bb$ agrees with $\bb$ outside their genes;
\item
$\prom(\tau\bb) = \prom(\bb)$;
\item \label{item:3}
if $\XX(\zz^\bb) = x_{i+1}^\ell\xx^\aa$ with $\ell = |\prom(\bb)|$, then
$\XX(\zz^{\tau\bb}) = x_{i+1}^\ell s_i(\xx^\aa)$; and
\item \label{item:4}
$\tau$ preserves column sums. In other words, if\/ $\bb' = \tau\bb$,
then $\sum_q b_{qp} = \sum_q b'{}_{\!qp}$ for any fixed column
index~$p$.
\end{thmlist}
In particular, $\XX(\ddem iw\zz^{\tau\bb}) = \dem i(x_{i+1}^\ell)
(s_i\xx^\aa)$.
\end{lemma}
\begin{remark} \label{rk:schur}
The squarefree monomials outside $J_w$ for a {Grassmannian
permutation}~$v$ (that is, a permutation having a unique descent) are in
natural bijection with the semistandard Young tableaux of the appropriate
shape and content. (This follows from Definition~\ref{defn:rc} and
Theorem~\ref{thm:rc}, below, along with the bijection in
\cite{KoganThesis} between rc-graphs and semistandard Young tableaux.)
Under this natural bijection, intron mutation
% transposition
reduces to an operation that arises in a well-known combinatorial proof
of the symmetry of the Schur function $\SS_v$ associated to~$v$.
\end{remark}
Our next proposition justifies the term `lifted Demazure operator' for
$\ddem iw$.
\begin{prop} \label{prop:lift}
Specializing $\zz$ to $\xx$ in $\ddem iw H(\cj w; \zz)$ yields $\dem i
H(\cj w; \xx)$. More generally, specializing $z_{qp}$ to $x_q/y_p$ in
$\ddem iw H(\cj w; \zz)$ yields $\dem i H(\cj w; \xx,\yy)$.
\end{prop}
\begin{proof}
Suppose $\zz^\bb \not\in J_w$ specializes to $\XX(\zz^\bb) =
x_{i+1}^\ell\xx^\aa$, where $\ell = |\prom(\bb)|$. The definition of
$\ddem iw\zz^\bb$ implies that
$$
\begin{array}{r@{\ \:=\ \:}l}
\XX(\ddem iw \zz^\bb)
&\sum_{d=0}^\ell x_i^d x_{i+1}^{\ell-d}\xx^\aa
\\[5pt] &\frac{x_{i+1}^{\ell+1} - x_i^{\ell+1}}{x_{i+1}-x_i}\xx^\aa
\\[5pt] &\dem i(x_{i+1}^\ell)\xx^\aa.
\end{array}
$$
If it happens that $s_i\xx^\aa = \xx^\aa$, so $\xx^\aa$ is symmetric in
$x_i$ and $x_{i+1}$, then
\begin{equation} \label{eq:fixed}
\XX(\ddem iw\zz^\bb)\ \:=\ \:\dem i(x_{i+1}^\ell) \xx^\aa\ \:=\ \:
\dem i(x_{i+1}^\ell \xx^\aa)\ \:=\ \:\dem i\XX(\zz^\bb).
\end{equation}
Of course, there will in general be lots of $\zz^\bb \not\in J_w$ whose
$\xx^\aa$ isn't fixed by $s_i$. We overcome this difficulty using
Lemma~\ref{lemma:involution}, which says how to pair each $\zz^\bb
\not\in J_w$ with a partner so that their corresponding $\XX \circ \ddem
iw$ sums add up nicely. Using the notation of the Lemma, notice that if
$\tau\bb = \bb$, then $s_i\xx^\aa = \xx^\aa$ and $\XX(\ddem iw\zz^\bb) =
\dem i\XX(\zz^\bb)$, as in~(\ref{eq:fixed}). On the other hand, if
$\tau\bb \neq \bb$, then the Lemma~implies
$$
\begin{array}{r@{\ \:=\ \:}l}
\XX(\ddem iw(\zz^\bb + \zz^{\tau\bb}))
&\dem i(x_{i+1}^\ell)(\xx^\aa + s_i\xx^\aa)
\\[5pt] &\dem i(x_{i+1}^\ell(\xx^\aa + s_i\xx^\aa))
\\[5pt] &\ddem iw(\XX(\zz^\bb + \zz^{\tau\bb}))
\end{array}
$$
because $\xx^\aa + s_i\xx^\aa$ is symmetric in $x_i$ and $x_{i+1}$. This
proves the $\ZZ^n$-graded statement.
The $\ZZ^{2n}$-graded version of the argument works mutatis mutandis by
the preservation of column sums under mutation (Definition~\ref{defn:mu}
and part~\ref{item:4} of Lemma~\ref{lemma:involution}), which allows us
to replace $\XX$ by $\XX_\YY$ and $\xx^\aa$ by a monomial in the $\xx$
variables and the inverses of the $\yy$ variables.
\end{proof}
Propopositions~\ref{prop:ev} and~\ref{prop:lift} imply the result used in
the proof of Theorem~\ref{thm:gb}.
\begin{cor} \label{cor:induction}
$H(\cj{ws_i}; \xx) = \dem i H(\cj w; \xx)$ if $\length(ws_i) <
\length(w)$. More generally, $H(\cj{ws_i}; \xx,\yy) = \dem i H(\cj w;
\xx,\yy)$ if $\length(ws_i) < \length(w)$.
\end{cor}
% \begin{remark}
% The `double' versions of Lemma~\ref{lemma:involution} and
% Proposition~\ref{prop:lift} hold, as well. Of course, the
% specialization $\XX$ must be replaced by the specialization $\psi$
% sending $z_{qp}$ either to $x_q/y_p$ or to $x_q y_p$. The double
% version of part~\ref{item:3} in Lemma~\ref{lemma:involution} reads:
% $$
% \hbox{\textit{If $\psi(\zz^\bb) = x_{i+1}^\ell\xx^\aa\yy^\cc$ then
% $\psi(\zz^{\tau\bb}) = x_{i+1}^\ell s_i(\xx^\aa)\yy^\cc$, where $\ell
% = |\prom(\bb)|$.}}
% $$
% The double version of the crucial last statement of
% Lemma~\ref{lemma:involution}, which now reads
% $$
% \hbox{\textit{In particular, $\psi(\ddem iw\zz^{\tau\bb}) = \dem
% i(x_{i+1}^\ell) (s_i\xx^\aa)\yy^\cc$,}}
% $$
% holds because {\em mutation and the involution $\tau$ do not change
% column sums}. Therefore the factor of $\yy^\cc$ carries through the
% entire proof of Proposition~\ref{prop:lift}. FALSE: THE INVOLUTION
% $\tau$ DOES, IN FACT, FIDDLE WITH COLUMN SUMS!
% \end{remark}
Before constructing this magic involution~$\tau$, we introduce some
necessary notation and provide examples.
% we need to dissect the exponent array $\bb$.
Recall that the union of rows $i$ and $i+1$ is the \textbf{gene} of $\bb$
(we view the row index~$i$ as being fixed for the discussion). Order the
boxes in columns east of $\start_i(\bb)$ in the gene of $\bb$ as in the
diagram, using the notation $\start_i(\bb)$ from~(\ref{eq:start}) on
page~\pageref{eq:start}:
% the end of Section~\ref{sub:mutation}.
$$
\begin{array}{l|c|c|c|c|c|c}
\multicolumn{1}{l}{\mbox{}}\\[-4ex]
\multicolumn{2}{l}{\mbox{}}&
\multicolumn{1}{c}{
\begin{array}{@{}c@{}}
\makebox[0pt]{$\scriptstyle \start_i(\bb)$}\\
\downarrow
\end{array}}
\\[.2ex]\cline{2-7}
\petit{i} &\toplinedots& 1 & 3 & 5 & 7 &\toplinedots\\\cline{3-6}
\petit{i+1}& & 2 & 4 & 6 & 8 & \\\cline{2-7}
\end{array}
$$
Now define five different kinds of blocks in the gene of $\bb$, called
the promoter, the start codon, exons, introns, and the stop codon.%
%
\footnote{All of these are terms from genetics. The DNA sequence
for a single gene is not necessarily contiguous. Instead, it
sometimes comes in blocks called \bem{exons}. The intervening
DNA sequences whose data are excised are called \bem{introns}
(note that the structure of the gene of an exponent array is
determined by its exons, not its introns). The \bem{promoter} is
a medium-length region somewhat before the gene that signals the
transcriptase enzyme where to attach to the DNA, so that it may
begin transcribing the DNA into RNA. The \bem{start codon} is a
short sequence signaling the beginning of the actual gene; the
\bem{stop codon} is a similar sequence signaling the end of the
gene.}
%
In the following, $k,\ell \in \NN$.
\begin{itemize}
\item
\bem{promoter}: the rectangle consisting of unnumbered boxes at the left
end
\vspace{-1.5ex}
\item
\bem{start codon}: the box numbered~$1$, which lies at
$(i,\start_i(\bb))$
\vspace{-1.5ex}
\item
\bem{stop codon}: the last numbered box, which lies at $(i+1,n)$
\vspace{-1.5ex}
\item
\bem{exon}: any sequence $2k,\ldots,2\ell+1$ (with $k \leq \ell$) of
consecutive boxes satisfying:
% all of the following:
\begin{thmlist}
\vspace{-2ex}
\item
the entries of~$\bb$ in the boxes corresponding to $2k,\ldots,2\ell+1$
are all zero;
\item
either box $2k+1$ is the start codon, or box~$2k$ has a nonzero entry
in~$\bb$;~and
\item
either box $2\ell$ is the stop codon, or box $2\ell+1$ has a nonzero
entry in~$\bb$
\end{thmlist}
\vspace{-2ex}
\item
\bem{intron}: any rectangle of consecutive boxes $2\ell+1,\ldots,2k$
(with $\ell < k$) satisfying:
% all of the following:
\begin{thmlist}
\vspace{-2ex}
\item
the rectangle contains no exons;
\item
box $2\ell+1$ is either the start codon or the last box in an exon; and
\item
box $2k$ is either the stop codon or the fisrt box in an exon
\end{thmlist}
\end{itemize}
Roughly speaking, the nonzero entries in gene$(\bb)$ are parititioned
into the promoter and introns, the latter being contiguous rectangles
having nonzero entries in their northwest and southeast corners. Exons
connect adjacent introns via bridges of zeros.
\begin{example} \label{ex:gene}
Suppose we're given a permutation $w$, an array $\bb$ such that $\zz^\bb
\not\in J_w$, and a row index $i$ such that $\start_i(\bb) = 4$ and $\bb$
has the gene in Figure~\ref{fig:intron}.
\begin{figure}[t]
\begin{rcgraph}
% 12 3 4 5
\begin{array}{rrc@{\ }c@{\hspace{-3.3ex}}c@{\hspace{-3.3ex}}c@{\hspace{-3.3ex}}
% 6 7 8
c@{\hspace{-3.3ex}}c@{\hspace{-3.3ex}}c@{\hspace{-3.3ex}}c}
&
{\rm promoter\colon}
&
\begin{array}{|c|c|c|}
\hline & \ \,&
\\\hline 2 & & 3
\\\hline
\end{array}\!\!
%
\\
\\
%
{\rm gene\ of\ }\bb\colon\hspace{-5ex}
&
\multicolumn{9}{r}{
% 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
\begin{array}{l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\multicolumn{1}{l}{\mbox{}}\\[-4ex]
\multicolumn{4}{l}{\mbox{}}&
\multicolumn{1}{c}{
\begin{array}{@{}c@{}}
\makebox[0pt]{start codon}\\
\downarrow
\end{array}}
\\\cline{2-21}
i & &\ \,& & 6& &\ \,& & & 4&\ \,& 3& 8& 6&\ \,& 2& & &\ \,&\ \,& 5
\\\cline{2-21}
i+1& 2& & 3& 1& 4& & 5& 3& 7& & & & & & 5& 1& 4& & &
\\\cline{2-21}
\multicolumn{20}{c}{}&
\multicolumn{1}{c}{
\begin{array}{@{}c@{}}
\uparrow\\
\makebox[0pt]{stop codon}
\end{array}}
\end{array}\:
}
%
\\
\\
%
&
{\rm exons\colon}
&&&
\raisebox{.15ex}{%
$
\begin{array}{|c|c|}
\cline{2-2}
\multicolumn{1}{c|}{} & 4
\\\hline 3
\\\cline{1-1}
\end{array}
$}
&&
\raisebox{-.15ex}{%
$
\begin{array}{|c|c|c|}
\cline{2-3}
\multicolumn{1}{c|}{} &\ \,& 3
\\\hline 7 &
\\\cline{1-2}
\end{array}
$}
&&
\begin{array}{|c|c|c|c|}
\cline{2-4}
\multicolumn{1}{c|}{} &\ \,& & 5
\\\hline 4 & &\ \,
\\\cline{1-3}
\end{array}\!
%
\\[3ex]
%
&
{\rm introns\colon}
&
&
\begin{array}{|c|c|c|c|c|}
\hline 6 & & & &
\\\hline 1 & 4 &\ \,& 5 & 3
\\\hline
\end{array}
&&
\begin{array}{|c|}
\hline 4
\\\hline 7
\\\hline
\end{array}
&&
\begin{array}{|c|c|c|c|c|c|c|}
\hline 3 & 8 & 6 &\ \,& 2 & &
\\\hline & & & & 5 & 1 & 4
\\\hline
\end{array}
&&
\begin{array}{|c|}
\hline 5
\\\hline
\\\hline
\end{array}
\end{array}
\end{rcgraph}
%Here's an intermediate step for mutation.
%\begin{rcgraph}
%% 12 3 4 5
%\begin{array}{rrc@{\ }c@{\hspace{-3.3ex}}c@{\hspace{-3.3ex}}c@{\hspace{-3.3ex}}
%% 6 7 8
% c@{\hspace{-3.3ex}}c@{\hspace{-3.3ex}}c@{\hspace{-3.3ex}}c}
%&
%{\rm promoter\colon}
%&
%\begin{array}{|c|c|c|}
% \hline & \ \,&
% \\\hline 2 & & 3
% \\\hline
%\end{array}\!\!
%%
%\\
%\\
%%
%{\rm gene\ of\ }\bb\colon\hspace{-5ex}
%&
%\multicolumn{9}{r}{
%% 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
% \begin{array}{l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
% \multicolumn{1}{l}{\mbox{}}\\[-4ex]
% \multicolumn{4}{l}{\mbox{}}&
% \multicolumn{1}{c}{
% \begin{array}{@{}c@{}}
% \makebox[0pt]{start codon}\\
% \downarrow
% \end{array}}
% \\\cline{2-21}
% i & &\ \,& & 6& &\ \,& & & 4&\ \,& 3& 8& 6&\ \,& 2& & &\ \,&\ \,& 5
% \\\cline{2-21}
% i+1& 2& & 3& 1& 4& & 5& 3& 7& & & & & & 5& 1& 4& & &
% \\\cline{2-21}
% \multicolumn{20}{c}{}&
% \multicolumn{1}{c}{
% \begin{array}{@{}c@{}}
% \uparrow\\
% \makebox[0pt]{stop codon}
% \end{array}}
% \end{array}\:
%}
%%
%\\
%\\
%%
%&
%{\rm exons\colon}
%&&&
%\raisebox{.15ex}{%
%$
%\begin{array}{|c|c|}
% \cline{2-2}
% \multicolumn{1}{c|}{} & 1
% \\\hline 1
% \\\cline{1-1}
%\end{array}
%$}
%&&
%\raisebox{-.15ex}{%
%$
%\begin{array}{|c|c|c|}
% \cline{2-3}
% \multicolumn{1}{c|}{} &\ \,& 1
% \\\hline 1 &
% \\\cline{1-2}
%\end{array}
%$}
%&&
%\begin{array}{|c|c|c|c|}
% \cline{2-4}
% \multicolumn{1}{c|}{} &\ \,& & 1
% \\\hline 1 & &\ \,
% \\\cline{1-3}
%\end{array}\!
%%
%\\[3ex]
%%
%&
%{\rm introns\colon}
%&
%&
%\begin{array}{|c|c|c|c|c|}
% \hline 6 & & & &
% \\\hline 1 & 4 &\ \,& 5 & 2
% \\\hline
%\end{array}
%&&
%\begin{array}{|c|}
% \hline 3
% \\\hline 6
% \\\hline
%\end{array}
%&&
%\begin{array}{|c|c|c|c|c|c|c|}
% \hline 2 & 8 & 6 &\ \,& 2 & &
% \\\hline & & & & 5 & 1 & 3
% \\\hline
%\end{array}
%&&
%\begin{array}{|c|}
% \hline 4
% \\\hline
% \\\hline
%\end{array}
%\end{array}
%\end{rcgraph}
For ease of comparison, we dissect here the mutated gene $\tau\bb$
of~$\bb$.
\begin{rcgraph}
% 12 3 4 5
\begin{array}{rrc@{\ }c@{\hspace{-3.3ex}}c@{\hspace{-3.3ex}}c@{\hspace{-3.3ex}}
% 6 7 8
c@{\hspace{-3.3ex}}c@{\hspace{-3.3ex}}c@{\hspace{-3.3ex}}c}
&
{\rm promoter\colon}
&
\begin{array}{|c|c|c|}
\hline & \ \,&
\\\hline 2 & & 3
\\\hline
\end{array}\!\!
%
\\
\\
%
{\rm gene\ of\ }\bb\colon\hspace{-5ex}
&
\multicolumn{9}{r}{
% 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
\begin{array}{l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\multicolumn{1}{l}{\mbox{}}\\[-4ex]
\multicolumn{4}{l}{\mbox{}}&
\multicolumn{1}{c}{
\begin{array}{@{}c@{}}
\makebox[0pt]{start codon}\\
\downarrow
\end{array}}
\\\cline{2-21}
i & &\ \,& & 7& 4&\ \,& 1& & 7&\ \,& 3& 7& &\ \,& & & &\ \,&\ \,& 1
\\\cline{2-21}
i+1& 2& & 3& & & & 4& 3& 4& & & 1& 6& & 7& 1& 4& & & 4
\\\cline{2-21}
\multicolumn{20}{c}{}&
\multicolumn{1}{c}{
\begin{array}{@{}c@{}}
\uparrow\\
\makebox[0pt]{stop codon}
\end{array}}
\end{array}\:
}
%
\\
\\
%
&
{\rm exons\colon}
&&&
\raisebox{.15ex}{%
$
\begin{array}{|c|c|}
\cline{2-2}
\multicolumn{1}{c|}{} & 7
\\\hline 3
\\\cline{1-1}
\end{array}
$}
&&
\raisebox{-.15ex}{%
$
\begin{array}{|c|c|c|}
\cline{2-3}
\multicolumn{1}{c|}{} &\ \,& 3
\\\hline 4 &
\\\cline{1-2}
\end{array}
$}
&&
\begin{array}{|c|c|c|c|}
\cline{2-4}
\multicolumn{1}{c|}{} &\ \,& & 1
\\\hline 4 & &\ \,
\\\cline{1-3}
\end{array}\!
%
\\[3ex]
%
&
{\rm introns\colon}
&
&
\begin{array}{|c|c|c|c|c|}
\hline 7 & 4 & & 1 &
\\\hline & &\ \,& 4 & 3
\\\hline
\end{array}
&&
\begin{array}{|c|}
\hline 7
\\\hline 4
\\\hline
\end{array}
&&
\begin{array}{|c|c|c|c|c|c|c|}
\hline 3 & 7 & &\ \,& & &
\\\hline & 1 & 6 & & 7 & 1 & 4
\\\hline
\end{array}
&&
\begin{array}{|c|}
\hline
\\\hline 4
\\\hline
\end{array}
\end{array}
\end{rcgraph}
\caption{Intron mutation}\label{fig:intron}
\end{figure}
The gene of $\bb$ breaks up into promoter, start codon, exons, introns,
and stop codon as indicated.
% The fact that the start codon has a nonzero entry is immaterial; we
% could have chosen it to be empty, but we'd still have to make this
% comment.
We will say something more about the mutated gene $\tau\bb$ in
Example~\ref{ex:tau}.
\end{example}
% The construction gets at the heart of what it takes to be in $J_w$: any
% antidiagonal $a \in J_w$ passing through rows $i$ and $i+1$ either
% skips one (or both) of them, in which case $a$ only cares which columns
% have at least one nonzero entry in row $i$ or $i+1$; or $a$ hits both
% rows, and $a$ only cares where the exons are.
If $\cc$ is an array having two rows filled with nonnegative
integers, then let
% \bem{inversion} of $\cc$ to be the
$\ol\cc$ be the rectangle obtained by rotating $\cc$ through an angle of
$180^\circ$. When $\cc$ is an intron, the rows of $\cc$ are identified
as rows $i$ and $i+1$ in a gene, and we view $\cc$ as an $n \times n$
array that happens to be zero outside of its $2 \times k$ rectangle.
\begin{defnlabeled}{Intron mutation} \label{defn:mutation}
Let $c_i$ and $c_{i+1}$
% respectively
be the sums of the entries in the top and bottom nonzero rows of an
intron~$\cc$, and set $d = |c_i-c_{i+1}|$.
% If $\cc$ contains neither the start codon nor the stop codon,
Then
\begin{eqnarray*}
\tau\cc &=& \left\{\begin{array}{@{}l@{\:\ \textrm{if}\ }l}
\ol{\mu^d(\ol\cc)} & c_i > c_{i+1} \\
\mu^d(\cc) & c_i < c_{i+1}
\end{array}\right.
\end{eqnarray*}
is the \bem{mutation} of~$\cc$.
% % If~$\cc$ contains the start or stop codon, then define $\tau\cc$ by
% % replacing $\mu^d$ with $\mu^{d-1}$ in the above equation.
% Given an exponent array $\bb$, let $\bb_+$ be obtained
% % respectively
% by adding~$1$ to the start and stop codons of~$\bb$. Similarly, let
% $\bb_-$ be obtained by subtracting~$1$.
Define the \bem{intron mutation} $\tau\bb$ of an exponent array $\bb$
% (for fixed $i$ and~$w$)
by
\begin{itemize}
\item
\vspace{-1.5ex}
adding~$1$ to the start and stop codons of~$\bb$;
\item
\vspace{-2ex}
mutating every intron in the gene of the resulting exponent array; and
then
\item
\vspace{-2ex}
subtracting~$1$ from the boxes that were the start and stop codons
of~$\bb$.
\end{itemize}
\end{defnlabeled}
% There can be at most one column with two nonzero entries, because {\em
% introns can't contain exons}. It is readily verified that a mutated
% % transposed
% intron contains no exons.
Intron mutation pushes the entries of each intron either upward from left
to right or downward from right to left---whichever initially brings the
row sums in that particular intron closer to agreement.
\begin{example} \label{ex:tau}
Although the ``look'' of $\bb$ in Example~\ref{ex:gene} completely
changes when it is mutated
% transposed
into $\tau\bb$, the columns of $\tau\bb$ containing a nonzero entry are
exactly the same as those in $\bb$, and the column sums are preserved.
Note that mutating
% transposing
the gene of $\tau\bb$ yields back the gene of $\bb$, {\em as long as
$\zz^\cc \not\in J_w$ and the location of the start codon hasn't
changed}. The proof of Lemma~\ref{lemma:involution} shows why $\tau$
always works this~way.%
\end{example}
\begin{lemma} \label{lemma:exons}
Intron mutation outputs an exponent array (that is, the entries are
nonnegative). Assume, for the purpose of defining exons in $\tau\bb$,
that the start codon of $\tau\bb$ lies at the same location as the start
codon in~$\bb$.
% The definition of `interior exon'---that is, an exon not containing the
% start codon---makes sense for $\tau\bb$.
% % , without reference to its start codon, which is undefined at this
% % stage, because we don't know yet that $\tau\bb \in \std(J_w)$.
% In fact,
% Intron mutation has been crafted specifically so the
% interior exons of $\tau\bb$ either coincide with interior exons of
% $\bb$, or have their southwest box in the promoter of~$\bb$
% % (and therefore have northeast box at the start codon of~$\bb$).
The boxes occupied by exons of $\tau\bb$ thus defined coincide with the
boxes occupied by exons of $\bb$ itself.
\end{lemma}
\begin{proof}
The definitions ensure that any intron not containing the start or stop
codon has nonzero northwest and southeast corners. After adding $1$ to
the start and stop codons, every intron has this property. Mutation of
such an intron leaves strictly positive entries in the northwest and
southeast corners (this is crucial---it explains why we have to add and
subtract the $1$'s from the codons), so subtracting~$1$ preserves
nonnegativity. Furthermore, intron mutation does not introduce any new
exons, because the nonzero entries in an intron both before and after
mutation follow a snake pattern that drops from row~$i$ to row~$i+1$
precisely once.%
\end{proof}
\begin{proofof}{Lemma~\ref{lemma:involution}}
First we show that $\tau\bb \in \std(J_w)$, or equivalently that
$\zz^{\tau\bb} \in J_w \implies \zz^\bb \in J_w$. Observe that $\zz^\bb
\in J_w$ if and only if $z_{ip}z_{i+1,n}\zz^\bb \in J_w$, where $p =
\start_i(\bb)$, by definition of $\start_i(\bb)$ and the fact that
$z_{i+1,n}$ is a nonzerodivisor modulo $J_w$ for all $w$. Therefore, it
suffices to demonstrate how an antidiagonal $a \in J_w$ dividing
$\zz^{\tau\bb}$ gives rise to a possibly different antidiagonal $a' \in
J_w$ dividing $z_{ip}z_{i+1,n}\zz^\bb$, where $p = \start_i(\bb)$. There
are five cases:
\begin{textlist}
\item $a$ intersects neither row $i$ nor row $i+1$;
\item the southwest variable in $a$ is in row $i$;
\item $a$ intersects row $i$ and continues south, but skips row $i+1$;
\item $a$ intersects both row $i$ and row $i+1$; or
\item $a$ skips row $i$ but intersects row $i+1$.
\end{textlist}
In each of these cases, $a'$ is constructed as follows. Outside the gene
of the generic matrix~$Z$, the new $a'$ will agree with $a$ in all five
cases, since $\bb$ and $\tau\bb$ agree outside of their genes. Inside
their genes, we may need some adjustments.
\begin{textlist}
\item Leave $a'=a$ as is.
\item Move the variable in row $i$ west to $z_{ip}$, using
Lemma~\ref{lemma:squish}(W).
\item \label{item:iii} The gene of $\bb$ has nonzero entries in precisely
the same columns as the gene of $\tau\bb$, by definition. Either $a$
already divides $z_{ip}\zz^\bb$, or moving the variable in row $i$ due
south to row $i+1$ yields $a'$ by Lemma~\ref{lemma:squish}(S).
\item
Use Example~\ref{ex:squish} to make $a'$ contain the nonzero entries
in some exon of~$\bb$ (see Lemma~\ref{lemma:exons}).
\item Same as~(\ref{item:iii}), except that either $a$ already divides
$z_{i+1,n}\zz^\bb$ or Lemma~\ref{lemma:squish}(N) says we can
move the variable due north from row $i+1$ to row $i$.
\end{textlist}
Now that we know $\tau\bb \in \std(J_w)$, we find that
\begin{eqnarray*}% \label{eq:crit}
\start_i(\tau\bb) &=& \start_i(\bb).
\end{eqnarray*}
Indeed, when $j \leq \start_i(\bb)$, we have $z_{ij}\zz^{\tau\bb} \in
J_w$ if and only if $z_{ij}\zz^\bb \in J_w$, because any antidiagonal
containing $z_{ij}$ interacts with $\tau\bb$ north of row $i$ and west of
column $\start_i(\bb)$, where $\tau\bb$ agrees with $\bb$. It follows
that $\prom(\tau\bb) = \prom(\bb)$,
% . Furthermore, granted the equality of promoters, $\tau$ is defined
% precisely so that
so $\tau\bb$ has the same exons as $\bb$ by Lemma~\ref{lemma:exons}.
% Since we have already seen in (\ref{eq:crit}) why $\tau\bb$ has the same
% start codon,
We conclude that $\tau\bb$ also has the same introns as $\bb$. The
statement $\tau^2 =$ identity holds because the partitions of the genes
of $\bb$ and~$\tau\bb$ into promoter and introns are the same, and
mutation
% transposition
on each of these blocks in the partition has order $1$ or $2$.
Part~\ref{item:3} follows intron by intron, except that the added and
subtracted $1$'s in the first and last introns cancel.%
\end{proofof}
\subsection{The initial complexes $\LL_w$: equidimensionality}%%%%%%%%%%%
\label{sub:Lw}%%%%%%%%%%%
\comment{AK will rewrite Example~\ref{ex:lex}.}
% \begin{defn} \label{defn:Lw}
Let $\LL_w$ be the Stanley-Reisner simplicial complex of~$J_w$, with
vertex set $[n]^2 = \{(q,p) \mid 1 \leq q,p \leq n\}$. That is, $\LL_w$
consists of the subsets of $[n]^2$ containing no antidiagonal in~$J_w$.
% \end{defn}
Faces of $\LL_w$ may be identified with coordinate subspaces in
$\spec(\cj w) \subseteq \mn$, as follows. Denote by $E_{qp}$ the
elementary matrix whose nonzero entry is in row~$q$ and column~$p$. We
identify the vertices with the variables $z_{qp}$ in the generic
matrix~$Z$, so that a coordinate subspace
\begin{equation} \label{eq:L}
L = \{z_{qp}\ \:=\ \:0 \mid (q,p) \in D_L \subseteq [n]^2\}\ \:=\ \:
{\rm Span}(E_{qp} \mid (q,p) \not\in D_L)
\end{equation}
is identified with the subset $[n]^2 \minus D_L$, when $L$ is being
regarded as a face of~$\LL_w$. We demonstrate here that the facets of
$\LL_w$ all have the same dimension.
The monomials $\zz^\bb$ that are nonzero in $\cj w$ are the so-called
\bem{standard monomials} for $J_w$, and are precisely those with support
sets
\begin{eqnarray*}
\supp(\zz^\bb) &:=& \{z_{qp} \in Z \mid z_{qp} \hbox{ divides }
\zz^\bb\}
\end{eqnarray*}
in the complex $\LL_w$. In particular, the maximal support sets of
standard monomials are the facets of $\LL_w$. (We will see in
Theorem~\ref{thm:rc} that the subsets $D_L \subset [n]^2$ for facets $L
\in \LL_w$ are the rc-graphs for the permutation $w$.) The next four
items (Example~\ref{ex:lex}--Lemma~\ref{lemma:lex}) serve as preparation
for the task of verifying that these facets all have the same
cardinality. (The next item isn't really an example of anything, but we
have to give it a number so we can refer to it.)
\begin{example} \label{ex:lex}
Suppose $v = w_0 s_{i_1}s_{i_2} \cdots s_{i_\ell}$, where $s_{i_1}s_{i_2}
\cdots s_{i_\ell}$ is the \bem{lexicographically first} reduced
expression for $w_0v$ in which $s_1 > s_2 > \cdots > s_{n-1}$. For the
identity permutation $v = 12345 \in S_5$ the lex first expression for
$w_0v$ is $s_1s_2s_1s_3s_2s_1s_4s_3s_2s_1$; and if $v = 31524$ then $w_0v
= s_2s_1s_3s_4s_3s_2$ is lex first. In general, one finds the lex first
expression for $w_0v$ by transforming the $n \times n$ antidiagonal
matrix $w_0^T$ into the permutation matrix $v^T$, beginning with sliding
the $1$ in column $n-1$ north into the correct position relative to the
$1$ in column $n$, then sliding the $1$ in column $n-2$ north into the
correct position relative to the $1$'s in columns $n$ and $n-1$, etc.
Because the $1$'s in columns to the west of the sliding $1$ are as far
south as possible, the entry of $\rk(w) = \rk(vs_{i_\ell})$ at position
$(i_\ell + 1,w(i_\ell + 1))$ is $1$.
\end{example}
\begin{lemma} \label{lemma:notafacet}
If $\zz^\bb$ divides $\zz^\cc \not\in J_w$ and $d \leq |\prom(\bb)|$,
then ${\mu_i}^d(\zz^\bb)$ divides ${\mu_i}^e(\zz^\cc)$ for some $e \leq
|\prom(\cc)|$. If $\supp(\zz^\bb) \in \LL_w$ is not a facet and $d \leq
|\prom(\bb)|$, then $\supp({\mu_i}^d(\zz^\bb)) \in \LL_{ws_i}$ is not a
facet.
\end{lemma}
\begin{proof}
If $\zz^\bb$ divides $\zz^\cc$, then $\start_i(\bb) \leq \start_i(\cc)$
by definition ($z_{ip}\zz^\bb \in J_w \implies z_{ip}\zz^\cc \in J_w$).
Therefore, if the $d^\th$ mutation of $\bb$ is the $k^\th$ occurring in
column $j < \start_i(\bb)$, we can choose $e$ so that the $e^\th$
mutation of $\cc$ is also the $k^\th$ occurring in column~$j$. If, in
addition, $z_{qp} \in \supp(\zz^\cc) \minus \supp(\zz^\bb)$, then either
$z_{qp}$ or $z_{q-1,p}$ ends up in $\supp({\mu_i}^e(\zz^\cc)) \minus
\supp({\mu_i}^d(\zz^\bb))$, depending on whether or not $(q,p) \in
\prom(\cc)$ and $p < j$. Note that ${\mu_i}^d(\zz^\bb)$ and
${\mu_i}^e(\zz^\cc)$ are not in $J_{ws_i}$ by Proposition~\ref{prop:ev},
so their supports are in $\LL_{ws_i}$.%
\end{proof}
\begin{lemma} \label{lemma:lex}
Suppose $w \in S_n$ and $\rk(w)_{i+1,w(i+1)} = 1$. Then:
\begin{thmlist}
\item \label{<}
$\length(ws_i) < \length(w)$; and
\item \label{zeros} $\zz^\bb \not\in J_w$ implies $b_{qp} = 0$ if $(q,p)
\leq (i+1,w(i+1)-1)$ or $(q,p) \leq (i,w(i)-1)$.
\end{thmlist}
Every permutation $v \in S_n$ aside from $w_0$ may be written as $v =
ws_i$ for some $w$ and~$i$ satisfying these hypotheses.
\end{lemma}
\begin{proof}
The condition on $\rk(w)$ implies part~\ref{<} because $(i,w(i))$ is
necessarily northeast of $(i+1,w(i+1))$. Part~\ref{zeros} is then
immediate from the definitions of $\rk(w)$ and $J_w$.
Example~\ref{ex:lex} implies the final statement.%
\end{proof}
\begin{claim} \label{claim:lex}
Assume the conditions of Lemma~\ref{lemma:lex}, and that $z_{i+1,w(i+1)}$
divides $\zz^\bb$ whenever $\supp(\zz^\bb) \in \LL_w$ is a facet. If
$\zz^\bb \not\in J_w$ has maximal support then $\west_{i+1}(\bb) = w(i+1)
< \start_i(\bb) = \west_i(\bb)$.
\end{claim}
\begin{proof}
Maximality of support and part~\ref{zeros} of Lemma~\ref{lemma:lex} imply
$\start_i(\bb) = \west_i(\bb)$. That $w(i+1) < \start_i(\bb)$ is then by
part~\ref{<} of Lemma~\ref{lemma:lex}. Part~\ref{zeros} of
Lemma~\ref{lemma:lex} implies $w(i+1) \leq \west_{i+1}(\bb)$, so $w(i+1)
= \west_{i+1}(\bb)$ by the hypothesis on $z_{i+1,w(i+1)}$.%
\end{proof}
\begin{lemma} \label{lemma:facet}
Under the conditions of Lemma~\ref{lemma:lex}, $z_{i+1,w(i+1)}$ divides
$\zz^\bb$ when $\supp(\zz^\bb) \in \LL_w$ is a facet.
\end{lemma}
\begin{proof}
Since this is obvious for $w = w_0$, we may prove it by downward
induction on $\length(w)$. More precisely, we need to show that if the
statement
\begin{eqnarray*}
\rk(w)_{k,w(k)} &=& 1\ \implies\ z_{k,w(k)}\in F\hbox{ for all facets }
F \in \LL_w
\end{eqnarray*}
is true, then the statement holds with $ws_i$ in place of $w$ when
$\rk(w)_{i+1,w(i+1)} = 1$.
By Proposition~\ref{prop:ev} and Lemma~\ref{lemma:notafacet}, every facet
of $\LL_{ws_i}$ is $\supp({\mu_i}^d(\zz^\bb))$ for some $\bb$ such that
$\supp(\zz^\bb)$ is a facet of $\LL_w$. Since mutation only affects rows
$i$ and $i+1$ (the gene of $\bb$), we find that $z_{k,w(k)} \in F$ for
all facets $F \in \LL_w$ implies $z_{k,w(k)} \in F$ for all facets $F \in
\LL_{ws_i}$, as long as $k \neq i,i+1$, in which case $w(k) = ws_i(k)$.
On the other hand, $ws_i(i+1) = w(i)$ and $\rk(ws_i)_{i+1,w(i)} \geq 2$
(the two $1$'s in the right-hand picture in (\ref{eq:w}) on
p.~\pageref{eq:w}), so we needn't worry about $k = i+1$. It remains to
show that $z_{i,w(i+1)} \in F$ for all facets $F \in \LL_{ws_i}$, given
the conclusion of the Lemma.
%since $ws_i(i) = w(i+1)$.
Let $\zz^\bb \not\in J_w$ have maximal support. By
Lemma~\ref{lemma:notafacet}, we may assume that \hbox{$b_{i+1,w(i+1)}
\geq 2$}. Observe that $\supp(\zz^\bb)$ is not itself a facet of
$\LL_{ws_i}$, because $\supp(\zz^\bb) \subsetneq \supp(\mu_i(\zz^\bb))$.
Therefore, every facet of $\LL_{ws_i}$ can be expressed as
$\supp({\mu_i}^d(\zz^\bb))$ for some {\em nonzero} $d \leq |\prom(\bb)|$,
and Claim~\ref{claim:lex} says that $z_{i,w(i+1)}$ divides all of
these.%
\end{proof}
If we knew a priori that $J_w$ were the initial ideal of $I(\ol X_w)$,
the following Proposition would follow from \cite{KSprimeIdeals}
(except for embedded components).
Since we are working in the opposite direction, we have to
prove it ourselves.
\begin{prop} \label{prop:pure}
The simplicial complex $\LL_w$ is pure, each facet having $\dim\ol X_w =
n^2 - \length(w)$ vertices; i.e.\ $\spec(\cj w)$ is equidimensional of
dimension $\dim\ol X_w$.
\end{prop}
\begin{proof}
$\LL_{w_0}$ has only one facet, with ${n+1 \choose 2} = n^2 - {n \choose
2}$ vertices $\{z_{qp} \in Z \mid q+p > n\}$, so we may again resort to
downward induction on $\length(w)$. In proving purity of $\LL_{ws_i}$
from that of $\LL_w$, we may assume by Lemma~\ref{lemma:lex} that $w$
and~$i$ satisfy the conditions put forth there.
Proposition~\ref{prop:ev} implies that the facets of $\LL_{ws_i}$ are
supports of monomials ${\mu_i}^d(\zz^\bb)$ for $\zz^\bb \not\in J_w$ and
$d \leq |\prom(\bb)|$. By Lemma~\ref{lemma:notafacet}, we may restrict
our attention to square monomials $\zz^{2\bb}$ with maximal support.
By definition, repeated mutation can only increase the support size of a
monomial by~$0$ or~$1$. It suffices therefore to demonstrate that if the
cardinalities of $\supp({\mu_i}^d(\zz^{2\bb}))$ and $\supp(\zz^{2\bb})$
are equal for some $d \leq |\prom(2\bb)|$, then there is some $e \leq
|\prom(2\bb)|$ such that $\supp({\mu_i}^d(\zz^{2\bb})) \subsetneq
\supp({\mu_i}^e(\zz^{2\bb}))$. If $d = 0$, then we may take $e = 1$ by
Lemma~\ref{lemma:facet}. If $d > 0$, then mutation will have just barely
pushed a row $i+1$ entry of $2\bb$ up to row $i$, and we may take $e =
d-1$. Containment requires the appropriate entry $b_{ij}$ to be $\geq
2$; strict containment is automatic, by checking cardinalities.%
\end{proof}
\begin{example} \label{ex:434}
The ideal $J_{1432}$ is generated by the antidiagonals of the five $2
\times 2$ minors contained in the union of the northwest $2 \times 3$ and
$3 \times 2$ submatrices of $(z_{ij})$:
\begin{eqnarray*}
J_{1432}
& = & \\\
& = & \ \cap \ \cap
\ \cap \ \cap
\.
\end{eqnarray*}
$\LL_{1432}$ is the join of a pentagon with a simplex having $11$
vertices $\{z_{11}\} \cup \{z_{rs} \mid r+s \geq 5\}$ (note $n=4$ here).
Each facet of $\LL_w$ therefore has $13 = 4^2 - \length(1432)$ vertices.%
\end{example}
We will have much more to say about the simplicial complexes $\LL_w$
throughout Section~\ref{sec:rc}, including better reasons why
Proposition~\ref{prop:pure} is true. In particular, the reader is
referred to Proposition~\ref{prop:mitosis}, Theorem~\ref{thm:rc}, and
Corollary~\ref{cor:shell}.
%\end{section}{Gr\"obner bases}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Applications of the Gr\"obner basis}%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:app}
Sections~\ref{sub:schub}--\ref{sub:groth} apply the Gr\"obner basis
Theorem~\ref{thm:gb} to Schubert and Grothendieck polynomials.
Sections~\ref{sub:minimal}--\ref{sub:other} relation of our Gr\"obner
basis to some others in the literature. Along the way,
Section~\ref{sub:minimal} contains a statement concerning minimal
Gr\"obner bases, and Section~\ref{sub:ladder} provides a closed formula
for the degrees of all one-sided ladder determinantal varieties.
% The reader interested in combinatorial applications of the methods and
% results of Section~\ref{sec:gb} is referred to Section~\ref{sec:rc}.
Finally, Section~\ref{sub:loci} outlines the connection between matrix
Schubert varieties and degeneracy loci for maps between vector bundles.
\subsection{Positive formulae for Schubert polynomials}\label{sub:schub}%
The original definition of Schubert polynomials was by Lascoux and
Sch\"utzenberger, via the divided difference operator recursion (our
Theorem~\ref{thm:oracle}; see also Corollary~\ref{notdefn:schubert}).
Since this formula involves negation, it is quite nonobvious from it that
the coefficients of $\SS_w(\xx)$ are in fact positive. This was first
proven in \cite{BJS, FSnilCoxeter} combinatorially, and
\cite{BSskewSchub,KoganThesis} geometrically.
We prove it now as an equation on multidegrees:
\begin{thm} \label{thm:positive}
The formula
\begin{eqnarray*}
[\ol X_w] &=& \sum_{L \in \LL_w} [L]
\end{eqnarray*}
holds for $\ZZ^n$-graded
% (left torus action)
and $\ZZ^{2n}$-graded
% ($T \times T^{-1}$ action)
multidegrees, where $[L]_{\ZZ^n} = \prod_{(q,p) \in D_L} x_p$, and
$[L]_{\ZZ^{2n}} = \prod_{(q,p) \in D_L} (x_p - y_q)$. For the
$\ZZ^n$-grading this is a formula for Schubert polynomials with positive
coefficients. For the $\ZZ^{2n}$-grading, it is a formula for double
Schubert polynomials with positive coefficients, if written in the
variables $\xx$ and~$-\yy$.
\end{thm}
\begin{proof}
Apply Theorem~\ref{thm:gb} to the second part of
Corollary~\ref{cor:grobdeg}: the multidegree of $\ol X_w$ in any grading
equals the multidegree of the zero set $\LL_w$ of~$J_w$. The specific
formulae for the multidegrees of coordinate subspaces is~(\ref{eq:wt}) in
Example~\ref{ex:subspace}, along with the specified weights earlier in
Section~\ref{sub:alex}.
\end{proof}
While it is nice to see the positivity demystified, this formula will
become much more effective when we have a better description of the
facets of $\LL_w$, in Section \ref{sec:rc}. At that point our formula
for Schubert polynomials will be seen to be equivalent to that in
\cite{BJS, FSnilCoxeter, BB}, while our formula for double Schubert
polynomials will follow from \cite[p.~134--5]{FKyangBax}. (This bypasses
the ``double rc-graphs'' of \cite{BB}.)
\begin{remark}\label{rem:effective1}
The version of this positivity in algebraic geometry is the notion of
``effective homology class'', meaning that it is representable by a
subscheme.
On the flag manifold, a homology class is effective exactly if it is a
nonnegative combination of Schubert classes. (Proof: one direction is a
tautology. For the other, if $X$ is a subscheme of the flag manifold, let
$B$ act on the Hilbert scheme containing the point $X$, and look at the
closure of the $B$-orbit through $X$. This will be projective because
the Hilbert scheme is, so by Borel's theorem there will be a fixed point,
necessarily a union of Schubert varieties, perhaps nonreduced.) In
particular the classes of monomials in the $x_i$ (the first Chern classes
of the standard line bundles; see Appendix~\ref{app:basics}) are not
usually effective.
We work instead on $\mn$, where a class is effective exactly if it is a
nonnegative combination of monomials. (Proof: instead of using $B$ to
degenerate a subscheme~$X$, use a $1$-parameter subgroup of the
$n^2$-dimensional torus. Algebraically, this exactly amounts to picking
a Gr\"obner basis.)
\end{remark}
\subsection{Grothendieck polynomials and $K$-theory}\label{sub:groth}%%%%
This subsection recovers a geometric result from our algebraic treatment
of matrix Schubert varieties: the Grothendieck polynomials represent the
$K$-classes of ordinary Schubert varieties in the flag manifold. We use
standard facts about flag variety and notions from equivariant algebraic
$K$-theory, on which background material can be found in
Appendix~\ref{app:K-theory} and Appendix~\ref{app:basics}.
Recall that the $K$-cohomology ring $K^\circ(\FL)$ is
the quotient of $\ZZ[\xx]$ by the ideal
\begin{eqnarray*}
K_n &=& \,
\end{eqnarray*}
where $e_d$ is the $d^\th$ elementary symmetric function. These
relations hold in $K^\circ(\FL)$ because the exterior power $\bigwedge^d
\kk^n$ of the trivial rank $n$ bundle is itself trivial, of rank $n
\choose d$. Furthermore, there can be no more relations, because
$\ZZ[\xx]/K_n$ is an abelian group of rank $n!\/$. Indeed, substituting
$\tilde x_k = 1 - x_k$, we find that $\ZZ[\xx]/K_n \cong
\ZZ[\tilde\xx]/\tilde K_n$, where $\tilde K_n = \$, and this quotient has rank $n!$ because it is the familiar
cohomology ring of $\FL$.
Thus it makes sense to say that a polynomial in $\ZZ[\xx]$ ``represents a
class'' in $K^\circ(\FL)$. Lascoux and Sch\"utzenberger, based on work
of Bernstein-Gel$'$fand-Gel$'$fand \cite{BGG} and Demazure \cite{Dem},
realized that the classes $[\OO_{X_w}] \in \ZZ[\xx]/K_n$ of (structure
sheaves of) Schubert varieties could be represented independently of~$n$.
To make a precise statement, let $\FLN = B \dom \glN$ be the manifold of
flags in $\kk^N$ for $N \geq n$, so $B$ is understood to consist of $N
\times N$ lower triangular matrices. Let $X_{w_N} \subseteq \FLN$ be the
Schubert variety for the permutation $w \in S_n$ considered as an element
of $S_N$ that fixes $n+1, \ldots, N$.
\begin{cor}[\cite{LSgrothVarDrap,LascouxGrothFest}] \label{cor:groth}
The Grothendieck polynomial $\GG_w(\xx)$ represents the class
$[\OO_{X_{w_N}}] \in K^\circ(\FLN)$ for all $N \geq n$.
\end{cor}
The main point is Theorem~\ref{thm:gb}, but we do still need a couple
more lemmas. Note that $\GG_w(\xx)$ is expressed without reference
to~$N$. Here is the reason why.
\begin{lemma} \label{lemma:stable}
The Grothendieck polynomial $\GG_w(\xx)$ in $n$ variables equals the
Grothen\-dieck polynomial $\GG_{w_N}(x_1,\ldots,x_N)$, whenever $w_N$
agrees with $w$ on $1,\ldots,n$ and fixes $n+1,\ldots,N$.
\end{lemma}
\begin{proof}
The ideal $I_{w_N}$ in the polynomial ring $\kk[z_{ij} \mid
i,j=1,\ldots,N]$ is extended from the ideal $I_w$ in the multigraded
polynomial subring~$\kk[\zz]$. Therefore $I_{w_N}$ has the same
multigraded Betti numbers as $I_w$, so their Hilbert numerators agree.
\end{proof}
\begin{lemma} \label{lemma:KB}
The natural map $K^\circ_B(\mn) \to K^\circ_T(\mn) \cong \ZZ[\xx^{\pm1}]$
is an isomorphism.
\end{lemma}
\begin{proof}
Suppose $E$ is a $B$-equivariant vector bundle on $\mn$. The global
sections $\Gamma E$ form a free $\ZZ^n$-graded $\kk[\zz]$-module because
$T \subset B$ (see Proposition~\ref{prop:kclass}). Any minimal generator
of $\Gamma E$ invariant under the unipotent matrices $N \subset B$
generates a $B$-submodule $\Gamma E'$ that is split as a $T$-submodule.
Since $[E]_B = [E']_B + [E/E']_B$, it follows by induction on rank that
$[E]_B$ is a sum of classes $[E']_B$ for line bundles $E'$. But
$B$-equivariant line bundles are the same as $T$-equivariant line
bundles, both being uniquely determined by the character of $T$ on the
$\kk$-vector space spanned by a generator of the global section module.
\end{proof}
\begin{proofof}{Corollary~\ref{cor:groth}}
In view of Lemma~\ref{lemma:stable}, we may as well assume $N = n$, since
$\GG_w$ doesn't care anyway. Let us justify the following diagram:
$$
\begin{array}{ccccc}
X_w
& &
& & \ol X_w
\\
\cap
& &
& & \cap
\\
B\dom\gln
&\twoheadleftarrow&\gln
&\hookrightarrow&\mn
\\[5pt]
K^\circ(B\dom\gln)
&\congto&K^\circ_B(\gln)
&\twoheadleftarrow& K^\circ_B(\mn)
\end{array}
$$
Pulling back vector bundles under the quotient map $B\dom\gln
\twoheadleftarrow \gln$ induces the isomorphism $K^\circ(B\dom\gln) \to
K^\circ_B(\gln)$. The inclusion $\gln \hookrightarrow \mn$ induces a
surjection $K^\circ_B(\gln) \twoheadleftarrow K^\circ_B(\mn)$ because the
classes of (structure sheaves of) algebraic cycles generate both of the
equivariant $K$-homology groups $K_\circ^B(\mn)$ and $K_\circ^B(\gln)$.
Now let $\wt X_w = \ol X_w \cap \gln$. Any $B$-equivariant resolution of
$\OO_{\ol X_w} = \ci w$ by vector bundles on $\mn$ pulls back to a
$B$-equivariant resolution $E_\spot$ of $\OO_{\wt X_w}$ on $\gln$.
Viewing a vector bundle on $\gln$ as a geometric object (i.e.\ as the
scheme $\SPEC(\sym^\spot\EE^\vee)$ rather than its sheaf of sections
$\EE$), the quotient $B \dom E_\spot$ is a resolution of $\OO_{X_w}$ by
vector bundles on $B\dom\gln$. Thus $[\OO_{\ol X_w}]_B \in
K^\circ_B(\mn)$ maps to $[\OO_{X_w}] \in K^\circ(B\dom\gln)$. The result
follows by identifying the $B$-equivariant class $[\OO_{\ol X_w}]_B$ as
the $T$-equivariant class $[\OO_{\ol X_w}]_T$ as in Lemma~\ref{lemma:KB},
and identifying the $T$-equivariant class as the Hilbert numerator
$\KK(\ci w;\xx) = \GG_w(\xx)$ by Proposition~\ref{prop:kclass} applied to
Theorem~\ref{thm:gb}.
\end{proofof}
\begin{remark}
The same line of reasoning recovers the double version of
Corollary~\ref{cor:groth}, in which $K^\circ(\FLN)$ is replaced by the
equivariant $K$-theory $K^\circ_{B^+}(\FLN)$ for the action of the
invertible upper triangular matrices on the right.
\end{remark}
\subsection{Minimal Gr\"obner bases and essential sets}\label{sub:minimal}
Our main purpose in the next three subsections is to relate our Gr\"obner
bases and connections with Schubert varieties to those of other authors.
Before we can say anything meaningful, however, we need some preliminary
results and notation. Therefore, this subsection and the next contain
some refinements and special cases of Theorem~\ref{thm:gb} that are
interesting in their own right, but also serve to facilitate the
comparative discussion in Section~\ref{sub:other}. Here, we determine
a minimal Gr\"obner basis for~$I_w$.
Define a \bem{one-sided ladder}%
%
\footnote{Most people recognize these as \bem{partitions}, but
apparently partitions occur too often in other guises in the
literature on determinantal ideals, so these had to be called
something else. Also, these are only special cases of more
general ladders that aren't partitions and have more the feel of
rungs.}
%
to be an order ideal in the poset $\ZZ_{>
0} \times \ZZ_{> 0}$---that is, a subset $\bL$ such that
$$
(q,p) \in \bL \implies (q',p') \in \bL \quad \hbox{for all } q' \leq q
\hbox{ and } p' \leq p.
$$
Although we usually drop the adjective `one-sided' in what follows, so
that `ladder' without qualification always means `one-sided ladder', it
is there to distinguish these ladders from the `two-sided ladders' that
appear in the literature on determinantal ideals. The \bem{southeast
corners} of a ladder $\bL$ are those locations $(q,p)$ such that
$(q',p')$ lies outside $\bL$ whenever $q' > q$ or $p' > p$. The
southeast corners are called \bem{outside corners} in the literature, but
often appear graphically as northeast corners due to reorientation of the
ladder.
The entries of the rank matrix $\rk(w)$ that are bounded above by~$r$
form a ladder. These \bem{rank ladders} are nested inside the $n \times
n$ array $[n]^2$ for increasing $r$:
$$
\Lambda_0(w) \subseteq \Lambda_1(w) \subseteq \cdots \subseteq
\Lambda_n(w) = [n]^2, \quad {\rm where} \quad \Lambda_r(w) = \{(q,p)
\mid \rk(w)_{qp} \leq r\}.
$$
Setting $\Lambda_{-1} = \nothing$ by convention, define the
\bem{essential set} of $w$ to be the set
\begin{eqnarray} \label{eq:ess}
\ess(w) &=& \bigcup_{r=0}^{n-1} \{\hbox{southeast corners } (q,p) \in
\Lambda_r \mid (q-1,p-1) \not\in \Lambda_{r-1}\}
\end{eqnarray}
of corners whose immediate northwest neighbor doesn't lie in a smaller
rank ladder.
\begin{example} \label{ex:ladder}
Let $w = 13865742$ as in Example~\ref{ex:intro}. The ladder
$\Lambda_0(w)$ is empty, whereas $\Lambda_1(w)$ and $\Lambda_2(w)$ are
depicted in Example~\ref{ex:intro}. The essential set of $w$ has four
elements: the single southwest corner $(7,2)$ of $\Lambda_1(w)$, and the
three southwest corners $(6,4)$, $(4,5)$, and $(3,7)$ of $\Lambda_2(w)$.
Observe that some southeast corners of $\Lambda_2(w)$ don't make it into
$\ess(w)$, and that none of the southeast of corners of $\Lambda_i(w)$
for $3 \leq i \leq 7$ make it in.
\end{example}
This alternate definition of essential set agrees with the original
defintion due to Fulton, because of the sentence after
\cite[Eq.~(3.8)]{FulDegLoc}, which in our language reads: ``A point
$(q,p)$ is in $\ess(w)$ if no entry $1$ in the permutation matrix $w^T$
lies due west or due north of or at $(q,p)$, and no entry $1$ in $w^T$
lies due east of or due south of or at $(q+1,p+1)$.'' The first
condition says that the entries of $\rk(w)$ in locations $(q-1,p)$,
$(q,p-1)$, and therefore $(q-1,p-1)$, are equal to the entry at $(q,p)$.
The second condition says that $(q,p)$ is a southeast corner of its rank
ladder. The reader is urged to consult Fulton's paper for numerous
examples, illustrations, and other ways of thinking about $\ess(w)$.
Before getting to the main result in this section, here is a preliminary
characterization of the minimal generators of the antidiagonal
ideal~$J_w$. Recall from Section~\ref{sub:mutation} what it means for a
rank condition to cause an antidiagonal. Here, we say that an
antidiagonal is caused by a subset $D \subseteq [n]^2$ if it is caused by
a rank condition $\rk(w)_{qp}$ for some $(q,p) \in D$.
\begin{lemma} \label{lemma:min}
An antidiagonal $a \in \kk[\zz]$ of degree $r+1$ is a minimal generator
of $J_w$ if and only if it is caused by $\Lambda_r$ but not by
$\Lambda_{r'}$ for $r' < r$.
\end{lemma}
\begin{proof}
The antidiagonal $a$ lies in~$J_w$ if and only if $a$ is caused by some
rank condition $\rk(w)_{qp}$. By definition, $\rk(w)_{qp} \leq r$, so
the remaining condition ensures that $a$ is not divisible by some smaller
antidiagonal in~$J_w$.
\end{proof}
Let $\ess(w)_r \subset [n]^2$ be the set of locations $(q,p) \in \ess(w)$
at which the rank matrix $\rk(w)$ has entry equal to~$r$.
\begin{prop} \label{prop:ess}
There is a unique set $A_w$ of minors in $I_w$ forming a minimal
Gr\"obner basis for every antidiagonal term order. The minors of size
\hbox{$r+1$} in $A_w$ are precisely those with antidiagonals caused by
$\ess(w)_r$ but not by $\ess(w)_{r'}$ for $r' < r$.
\end{prop}
\begin{proof}
The uniqueness of a minimal Gr\"obner basis consisting of minors follows
from Theorem~\ref{thm:gb} and the fact that a minor is determined by its
antidiagonal. If $\rk(w)_{qp}$ causes a degree $r+1$ antidiagonal $a$
but $(q,p) \not\in \ess(w)_r$, then chopping off the northeast or
southwest end of $a$ yields an antidiagonal caused by $\Lambda_{r-1}(w)$
and dividing~$a$. Therefore every minimal generator of $J_w$ is caused
by $\ess(w)$.
When $a \in J_w$ is a minimal generator, the criterion of
Lemma~\ref{lemma:min} clearly implies that $\ess(w)_{r'} \subseteq
\Lambda_{r'}(w)$ can't cause $a$ for $r' < r$. On the other hand, if $a
\in J_w$ is not a minimal generator, then $\Lambda_{r'}(w)$ causes $a$
for some $r' < r$ by Lemma~\ref{lemma:min}. By definition, $|a \cap
Z\sub qp| > r'$ for some $(q,p) \in \Lambda_{r'}(w)$. Some minimal
generator of $J_w$ divides the antidiagonal $a \cap Z\sub qp$, and this
minimal generator is caused by $\ess(w)_{r''}$ for some $r'' \leq r'$.
Therefore $\ess(w)_{r''}$ causes $a \cap Z\sub qp$, and hence $a$ as
well.
\end{proof}
\begin{example} \label{ex:ess}
Although the $165$ minors of size $2$ and~$3$ in Example~\ref{ex:intro}
do form a Gr\"obner basis, the minimal Gr\"obner basis $A_w$ has only
$102$ elements. It consists of the $21$ minors of size~$2$
in~$\Lambda_1(w)$ and
%
\comment{
$$
[2{6 \choose 3}] + [(2 \cdot 2 \cdot 1+1 \cdot 1 \cdot 1){4 \choose 3}]
+[(2 \cdot 3 \cdot 2+1 \cdot 3 \cdot 1+1 \cdot 3 \cdot 2){3 \choose 3}]
= 2 \cdot 20 + 5 \cdot 4 + (12+3+6) = 40 + 20 + 21 = 81
$$}%
%
$81$ of the minors of size~$3$ in~$\Lambda_2(w)$.
\end{example}
The minimal Gr\"obner basis $A_w$ in Proposition~\ref{prop:ess} is rarely
a reduced Gr\"obner basis, because a minimal generator $a \in J_w$ can
divide a monomial $m \neq a$ in some minor in~$A_w$.
\begin{example} \label{ex:13254}
Label each point in the essential set of $w = 13254 \in S_5$ by $*$, so
that
$$
%\begin{array}{@{}c@{\ \mapsto\ }c@{}}
%1 & 1\\
%2 & 3\\
%3 & 2\\
%4 & 5\\
%5 & 4
%\end{array}\,,\ \
w^T =
\begin{array}{|c|c|c|c|c|}
\hline
1&&&&\\\hline
&*&1&&\\\hline
&1&&&\\\hline
&&&*&1\\\hline
&&&1&\\\hline
\end{array}\,,
\ \ \hbox{and} \quad
A_w =
\left\{
\left|\begin{array}{cc}z_{11}&z_{12}\\z_{21}&z_{22}\end{array}\right|,
\left|\begin{array}{cccc}z_{11}&z_{12}&z_{13}&z_{14}\\
z_{21}&z_{22}&z_{23}&z_{24}\\
z_{31}&z_{32}&z_{33}&z_{34}\\
z_{41}&z_{42}&z_{43}&z_{44}\end{array}\right|
\right\}
$$
consists of $\det(Z\sub 22)$ and $\det(Z\sub 44)$. Although these two
minors happen to generate $I_w$ minimally,
% In general, the way to find the essential set is to cross out all boxes
% due south or due east of a $1$, and then choose the remaining empty
% boxes from which moving due south or due east lands in a crossed-out
% box. Many more examples can be found in \cite{FulDegLoc}.
the minimal Gr\"obner basis consists of $\det(Z\sub 22)$ along with the
non-minor $\det(Z\sub 44) - \det(Z\sub 22) \cdot
(z_{33}z_{44}-z_{34}z_{43})$.
For the record, observe that the minors in $I_{13254}$ never form a
Gr\"obner basis for a diagonal term order (in which the leading term of
every minor is its diagonal term), because the main diagonal of $Z\sub
22$ divides that of $Z\sub 44$. In contrast, the antidiagonal terms are
relatively prime, so the Gr\"obner basis condition is immediate.%
\end{example}
For later reference, here is the reason why Fulton introduced the
essential set in the first place \cite[Lemma~3.10(a)]{FulDegLoc}.
\begin{cor} \label{cor:ess}
The minors of size $1+\rk(w)_{qp}$ for $(q,p) \in \ess(w)$
generate~$I_w$.
\end{cor}
\subsection{Degrees of ladder determinantal varieties}\label{sub:ladder}%
We continue our preparation for comparison to the literature in the next
subsection with a formula for the degrees of some popular determinantal
ideals. A modicum of comparison of various authors' work also occurs
here.
Let $\bL$ be a ladder, and regard $\bL$ as a subset of a sufficiently
large (to be made more precise in Proposition~\ref{prop:vex}) generic $n
\times n$ matrix $Z$ of variables $z_{ij}$. The boundary of $\bL$ is a
ribbon strip proceeding from the western edge of the square to its
northern edge. Consider a sequence of boxes $(b_1,a_1),\ldots,(b_k,a_k)$
lying along the boundary of~$\bL$, so that
\begin{equation} \label{eq:boxes}
a_1 \leq a_2 \leq \cdots \leq a_k \quad\hbox{and}\quad
b_1 \geq b_2 \geq \cdots \geq b_k.
\end{equation}
Fill the boxes $(b_\ell,a_\ell)$ with nonnegative integers $r_\ell$
satisfying
\begin{equation} \label{eq:ranks}
0 < a_1 - r_1 < a_2 - r_2 < \cdots < a_k - r_k \quad\hbox{and}\quad
b_1 - r_1 > b_2 - r_2 > \cdots > b_k - r_k > 0.
\end{equation}
Define the \bem{ladder determinantal ideal} $I(\ub a, \ub b, \ub r)$ to
be generated by the minors of size $r_\ell$ in the northwest $a_\ell
\times b_\ell$ corner of $Z$ for all $\ell \in 1,\ldots,k$.
Since `ladders' are just another name for `partitions' one might also
call these `partition determinantal ideals', but as we shall argue, the
better name is actually `vexillary determinantal ideals'. Indeed, for
reasons that will become clear shortly, we refer to the data $(\ub a,\ub
b,\ub r)$ as a \bem{vexillary essential set}. The condition
in~(\ref{eq:ranks}) on the vexillary essential set simply ensures that
the vanishing of the minors of size $r_\ell$ in the northwest $b_\ell
\times a_\ell$ submatrix does not imply the vanishing of the minors of
size $r_{\ell'}$ in the northwest $b_{\ell'} \times a_{\ell'}$ submatrix
when $\ell \neq \ell'$.
The polynomials generating $I(\ub a,\ub b,\ub r)$ involve only the
variables inside $\bL$, so that $I(\ub a,\ub b,\ub r)$ could be
considered as an ideal in the polynomial ring $\kk[z_{ij} \mid (i,j) \in
\bL]$. There is little harm, however, in allowing extra variables
outside the ladder; nothing substantial about the algebra (free
resolutions, homology) changes, and there results only a product with an
affine space at the level of varieties.
% \begin{example} \label{ex:vexess}
% {bf Here is a relatively large (but small-printed) example.}
% $$
% {\rm PICTURE\ HERE}
% $$
% \end{example}
The ladder determinantal ideals in \cite{GonLakLadDetSchub} have
vexillary essential sets whose ranks~$\ub r$ weakly increase from
southwest to northeast (in our language). In contrast, those treated in
\cite{GoncMilMixLadDet} have vexillary essential sets with no two boxes
in the same row or column---that is, with strict inequalities
in~(\ref{eq:boxes})---but no restriction other than~(\ref{eq:ranks}) on
the numbers $\ub r$ in those boxes. Gonciulea and C.\thinspace{}Miller
in fact mention that arbitrary vexillary essential sets can be treated
with their methods \cite[p.~106--107]{GoncMilMixLadDet}. Fulton
considered many aspects of ladder determinantal ideals in the generality
presented here, \cite[Section~9]{FulDegLoc}. The connection is through
\bem{vexillary permutations} (also known as \bem{$2143$-avoiding} and
\bem{single-shaped} permutations), and Fulton's Proposition~9.6:
\begin{prop}[\cite{FulDegLoc}] \label{prop:vex}
Given a vexillary essential set $(\ub a,\ub b,\ub r)$, there exists an
integer $n_0 \leq a_k + b_1$ such that for all $n \geq n_0$, a unique
permutation $w \in S_n$ satisfies
$$
\ess(w) = \{(b_1,a_1),(b_2,a_2),\ldots,(b_k,a_k)\} \quad\hbox{and}\quad
\rk(w)_{b_\ell a_\ell} = r_\ell \quad\hbox{for } 1 \leq \ell \leq k.
$$
The permutation $w$ is vexillary, and every vexillary permutation has a
vexillary essential set. The length of $w$, which equals the codimension
of $I(\ub a,\ub b,\ub r)$, is $m_1 p_1 + m_2 p_2 + \ldots + m_k p_k$,
where the integers $p_1,\ldots,p_k$ and $m_1,\ldots,m_k$ are defined by
\begin{equation} \label{eq:pm}
\begin{array}{@{}c@{}}
p_1 = a_k - r_k,\ p_2 = a_{k-1} - r_{k-1},\ldots,p_k = a_1 - r_1 \quad
\hbox{and}\\[2ex]
m_1 = b_k - r_k,\ m_2 = (b_{k-1}-r_{k-1}) - (b_k-r_k),\ldots,m_k =
(b_1-r_1) - (b_2 - r_2).
\end{array}
\end{equation}
\end{prop}
Briefly then, Proposition~\ref{prop:vex} along with
Corollary~\ref{cor:ess} says that the class of one-sided ladder
determinantal ideals $I(\ub a,\ub b,\ub r)$ coincides with the class of
vexillary Schubert determinantal ideals~$I_w$, and provides their
codimensions. See Fulton's paper and \cite[Chapter~1]{NoteSchubPoly} for
details concerning vexillary permutations, and ways of visualizing the
numbers $p_\ell$ and~$m_\ell$. These numbers appear implicitly in the
codimension formulae of \cite{GonLakLadDetSchub} and
\cite{GoncMilMixLadDet}. Note (as Fulton does) that the probability of a
permutation being vexillary decreases exponentially to zero as $n$
approaches infinity, so the class of Schubert determinantal ideals is
significantly more general than ladder determinantal ideals. In fact,
\cite[Section~3]{FulDegLoc} says that Schubert determinantal ideals are
the largest possible class of prime determinantal ideals defined by rank
conditions of the form $\rank(Z\sub qp) \leq r_{qp}$.
Now we provide a formula for the degree of a ladder determinantal ideal.
\begin{thm} \label{thm:vexdeg}
%[Formula Ni\c{c}oise\comment{spelling?}]
Let $(\ub a,\ub b,\ub r)$ be a vexillary essential set and $\lambda =
(p_1^{m_1},p_2^{m_2},\ldots,p_k^{m_k})$ be the partition determined by
the integers from (\ref{eq:pm}), so that
\begin{eqnarray} \label{eq:lambda}
\lambda &=& (\underbrace{p_1,\ldots,p_1}_{m_1}\ ,\
\underbrace{p_2,\ldots,p_2}_{m_2}\ ,\ \ldots\ ,\
\underbrace{p_k,\ldots,p_k}_{m_k}).
\end{eqnarray}
Given $q \leq m_1 + m_2 + \ldots + m_k$, define $\ell(q)$ by $\lambda_q =
p_{\ell(q)}$, so that we have $\ell(q) = \min\{\ell \mid q \leq m_1 +
\ldots + m_\ell\}$. The degree of the ladder determinantal ideal $I(\ub
a,\ub b,\ub r)$ is
\begin{eqnarray*}
\deg(I(\ub a,\ub b,\ub r)) &=& \det\left[{b_{\ell(q)} + \lambda_q - q + p
- 1 \choose b_{\ell(q)} - 1}\right]_{q,p=1}^{m_1+\cdots+m_k}.
\end{eqnarray*}
\end{thm}
\begin{proof}
With $w$ as in Proposition~\ref{prop:vex} and $\lambda$ as
in~(\ref{eq:lambda}), the double Schubert polynomial $\SS_w(\xx,\yy)$ is
the multi-Schur polynomial \cite[Eq.~(9.4)]{FulDegLoc} because $w$ is
vexillary; this is \cite[Proposition~9.6(d)]{FulDegLoc}. Setting the $y$
variables to $0$ and the $x$ variables to~$1$ in this determinantal
expression yields the desired formula, by
Proposition~\ref{prop:coarsen}.%
\end{proof}
The proof of Theorem~\ref{thm:vexdeg} actually points out a much stronger
statement: the \hbox{$\ZZ^{2n}$-graded} multidegree of a vexillary
determinantal ideal is a multi-Schur polynomial, which has an explicit
determinantal expression. (Those wishing to see the determinantal
expression in its full glory can check \cite{FulDegLoc} for a brief
introduction, or \cite{NoteSchubPoly} for much more.) It would be
desirable to make the Hilbert series in Theorem~\ref{thm:gb} just as
explicit. Is there an analogously ``nice'' formula%
%
\footnote{In the introductions to \cite{AbhyankarECYT} and its
second chapter, Abhyankar writes of formulae he first presented
at a conference at the University of Nice, in France. Although
his formulae enumerate certain kinds of tableaux, his results
were used to obtain formulae for degrees and Hilbert series of
determinantal ideals. Authors since then have been looking for
``nice'' (uncapitalized, and always in quotes) formulae for
Hilbert series of determinantal ideals; cf.\
\cite[p.~3]{HerzogTrung} and \cite[p.~55]{AbhyanKulkarHilb}.}
%
for vexillary double Grothendieck polynomials $\GG_w(\xx,\yy)$, or an
ordinary version for $\GG_w(\xx)$, or even for $\GG_w(t,\ldots,t)$?
\subsection{Between Schubert varieties and determinantal ideals:\\
comparison with known results}\label{sub:other}%%
Broadly speaking, the study of generic determinantal ideals has
progressed in recent years well beyond its classical origins, which
concerned ideals generated by equal size minors in rectangular matrices
and subvarieties of Grassmannians. Recent trends involve collections of
variously sized minors in more general types of regions inside generic
matrices, all of which go under the name `ladder' with some prepended
adjectives, such as `one-sided', `two-sided', `one-corner', or `wide'
(there are many others). The varieties defined by each class of ladder
determinantal ideals come with some connection to Schubert varieties in
type~$A$ partial flag manifolds, frequently via opposite big cells.
Although we review the bare necessities below, we refer the reader to
Gonciulea and C.\thinspace{}Miller's paper \cite{GoncMilMixLadDet} for a
well-written and nearly self-contained (as noted in Mathematical Reviews,
MR~2001d:14055) entry point into the literature on this popular subject.
In particular, their Introduction contains a brief historical account.
Let $\bL \subset [n]^2$ be a ladder (as before, `ladder' without
qualification here always means `one-sided ladder'). Define $\AL$ to be
the vector subspace of the $n \times n$ matrices $\mn$ consisting of
matrices whose nonzero entries lie in $\bL$, and let $w_0^T+\AL$ be the
translate of the subspace $\AL$ by the unit antidiagonal matrix~$w_0^T$.
If $\bL$ does not intersect the main antidiagonal of $[n]^2$, so that
$\bL$ lies strictly within the upper-left triangle, then $w_0^T+\AL$ is
an affine subspace of $\mn$ that happens to be contained in~$\gln$. In
other words, $w_0^T+\AL$ is contained in the subset $w_0^T N$ of $\gln$,
where $N$ is the subgroup of $B$ (the {\em lower} triangular Borel group)
having $1$'s along the main diagonal. The set $w_0^T N$ consists of the
matrices with $1$'s along the main antidiagonal and zeros below it.
Since $w_0^T N$ intersects $Bw_0^T$ precisely at the long word $w_0^T$
in~$\gln$, the variety $w_0^T N$ (and hence also $w_0^T+\AL$) maps
isomorphically to its image in $B\dom\gln$, which is called the
\bem{opposite big cell} $\Omega^\circ$ in the full flag variety.
% The preimage in $\gln$ of the opposite big cell in $B\dom\gln$ is the
% double coset $Bw_0^TB$ (not $Bw_0^TB_+$).
We summarize this discussion in a lemma.
\begin{lemma} \label{lemma:AL}
If $\bL$ is a ladder that does not intersect the main antidiagonal, then
$(w_0^T+\AL) \times \AA_\bL \subset \gln$ maps isomorphically to the
opposite big cell $\Omega^\circ$ in the full flag variety $B\dom\gln$,
where $\AA_\bL$ is the vector space of matrices whose nonzero entries lie
strictly above the main antidiagonal but outside of\/~$\bL$.
\end{lemma}
Given $w \in S_n$, let $\bL = \bL(w)$ be the smallest ladder in $[n]^2$
containing the essential set $\ess(w)$, and set $\kk[\bL] := \kk[z_{ij}
\mid (i,j) \in \bL]$, the coordinate ring of $\AL \cong w_0^T+\AL$. By
Corollary~\ref{cor:ess}, the ideal $\ol I_w := I_w \cap \kk[\bL]$ cuts
out a variety in $\AL \cong w_0^T+\AL$ whose product with
$\AA^{[n]^2\minus\bL}$ coincides with $\ol X_w$, where
$\AA^{[n]^2\minus\bL}$ is the vector space of matrices in $\mn$ having
zeros inside~$\bL$. In other words, $\ol I_w \kk[\zz] = I_w$, so it is
natural to consider $\ol I_w$ instead of~$I_w \subseteq \kk[\zz]$. There
exist in the literature several different but closely related ways to
pass from the Schubert variety $X_w \subseteq B\dom\gln$ to the
determinantal ideal $\ol I_w$. All of them first involve taking the
preimage $\wt X_w \subseteq \gln$ of $X_w$.
\begin{prop} \label{prop:four}
Given $w \in S_n$ and the determinantal locus $\wt X_w \subseteq \gln$,
assume that $\bL \supseteq \bL(w)$ misses the main antidiagonal in
$[n]^2$. The ideals of the varieties obtained by the following four
procedures all coincide with $\ol I_w \subseteq \kk[\bL]$:
\begin{thmlist}
\item \label{ful}
Project $\wt X_w$ to $\AL$.
\item \label{goncmil}
Intersect $\wt X_w$ with $w_0^T+\AL$.
\item \label{opp}
Intersect $\wt X_w$ with $Bw_0^TB$ and then project to $w_0^T+\AL$.
\item \label{km}
Take the closure of $\wt X_w$ inside of $\mn$, and then project~to~$\AL$.
\end{thmlist}
\end{prop}
\begin{proof}
Procedure~\ref{km} yields $\ol I_w$ by definition.
The ideal of $\wt X_w$ is the extension $\ol I_w \kk[\zz][\det^{-1}]$ of
$\ol I_w$ from $\kk[\bL]$. Procedure~\ref{goncmil} results algebraically
from the map $\kk[\zz][\det^{-1}] \onto \kk[\bL]$ sending $z_{ij}$ to~$0$
whenever $i + j \neq n+1$ and $(i,j) \not\in \bL$, and sending
$z_{i,n+1-i}$ to~$1$ for all~$i$. Since none of the generators of $\ol
I_w$ involve these variables, the image of $\ol I_w \kk[\zz][\det^{-1}]$
in $\kk[\bL]$ is again~$\ol I_w$.
Due to the stability of both $\wt X_w$ and $Bw_0^TB$ under multiplication
by $B$ on the left, the intersection in procedure~\ref{opp} commutes with
projection to~$B\dom\gln$. Procedures \ref{goncmil} and~\ref{opp}
therefore yield the same result by Lemma~\ref{lemma:AL}, because
$Bw_0^TB$ is the preimage in $\gln$ of the opposite big
cell~$\Omega^\circ$.
Finally, the variety produced by procedure~\ref{ful} is contained in the
variety produced by procedure~\ref{km}, but contains the variety produced
by procedure~\ref{opp}.
\end{proof}
Of the four procedures in Proposition~\ref{prop:four}, only the last
works even when $\bL$ intersects the main antidiagonal. We postpone the
discussion of the origins of these procedures until later in this
subsection. First, we use the equivalence to express matrix Schubert
varieties in $\mn$ as open subsets of honest Schubert varieties in higher
dimensional flag varieties.
%o o o o o o o . . . . . . 1 n = 7
%o o o o o o o . . . . . 1 . 2n = 14
%o o o o o o o . . . . 1 . . there are 2(7 choose 2) = 7.6 new variables
%o o o o o o o . . . 1 . . . = n(n-1)
%o o o o o o o . . 1 . . . .
%o o o o o o o . 1 . . . . .
%o o o o o o o 1 . . . . . .
%. . . . . . 1 . . . . . . .
%. . . . . 1 . . . . . . . .
%. . . . 1 . . . . . . . . .
%. . . 1 . . . . . . . . . .
%. . 1 . . . . . . . . . . .
%. 1 . . . . . . . . . . . .
%1 . . . . . . . . . . . . .
\begin{cor} \label{cor:bigcell}
Given $w \in S_n$, consider $w$ as an element of $S_{2n}$ fixing each of
\hbox{$n+1,\ldots,2n$}. The product $\ol X_w \times \kk^{n^2-n}$ of the
matrix Schubert variety $\ol X_w \subseteq \mn$ with a vector space of
dimension $n^2-n$ is isomorphic to the intersection of the Schubert
variety $X_{w,2n} \subseteq B\dom{G_{\!}L_{2n}}$ with the opposite big
cell in $B\dom{G_{\!}L_{2n}}$.
\end{cor}
\begin{proof}
Replace $n$ by $2n$ in Lemma~\ref{lemma:AL} and
Proposition~\ref{prop:four}, and take $\bL$ to be the whole square
$[n]^2$ considered as a ladder in $[2n]^2$. The point is that the
northwest $n \times n$ submatrix of a $2n \times 2n$ matrix misses the
main antidiagonal, and leaves
% The space $\AA_\bL$ in Lemma~\ref{lemma:AL} has dimension $2 \cdot {n
% \choose 2} = n^2-n$.
$2 \cdot {n \choose 2} = n^2-n$ empty spots above the main antidiagonal.
\end{proof}
In fact one can do somewhat better than the corollary, if one is only
interested in the commutative algebra of the determinants
generating~$I_w$: the product of the zero set of $\ol I_w$ with some
vector space is isomorphic to the intersection of the Schubert variety
$X_{w,N}$ in $B\dom\glN$ with the opposite big cell in $B\dom\glN$, where
we choose $N = \max\{q+p \mid (q,p) \in \ess(w)\}$, so that $N < 2n$.
The point of the above results is a rather general equivalence principle
that has been applied many times in the literature. By a \bem{local
condition}, we mean a condition that holds for a variety whenever it
holds on each subvariety in some open cover.
\begin{thm} \label{thm:equiv}
Let $\mathfrak C$ be a local condition that holds for a variety $X$
whenever it holds for the product of $X$ with any vector space. Then
$\mathfrak C$ holds for every Schubert variety in every flag variety if
and only if $\mathfrak C$ holds for all matrix Schubert varieties.
\end{thm}
\begin{proof}
If $\mathfrak C$ holds for Schubert varieties, then it holds for matrix
Schubert varieties by Corollary~\ref{cor:bigcell}. On the other hand, if
$\mathfrak C$ holds for the matrix Schubert variety $\ol X_w \subseteq
\mn$, then it holds for $\wt X_w = \ol X_w \cap \gln$. Therefore
$\mathfrak C$ holds for the Schubert variety $X_w \subseteq B\dom\gln$,
because $\wt X_w$ is locally isomorphic to the product of $X_w$ with~$B$,
which is an open subset of a vector space.
\end{proof}
The local conditions that are of primary interest include normality, the
Cohen--Macaulay property, and rational singularities. Since all of these
are known for Schubert varieties \cite{ramanathanCM,RamRamNormSchub} and
can be deduced for Schubert determinantal ideals using an argument of
Fulton \cite[Section~3]{FulDegLoc}, our inclusion Theorem~\ref{thm:equiv}
is mostly for the record; we know of no new facts that can be derived
from it. On the other hand, new proofs become available:
Section~\ref{sub:cm} derives Cohen--Macaulayness of Schubert varieties by
working directly with initial ideals of Schubert determinantal ideals.
We purposely avoid deriving facts about Schubert determinantal ideals
from the corresponding facts about Schubert varieties, to keep this
exposition self-contained.
Procedure~\ref{ful} in Proposition~\ref{prop:four} is Fulton's
construction that appears in \cite[Section~3]{FulDegLoc}. His argument
there essentially proves the `only if' part of Theorem~\ref{thm:equiv}.
The idea behind procedure~\ref{goncmil} seems to be due originally to
Mulay \cite{MulayLadders}, and has since then been used explicitly only
in the ladder determinantal case (as far as we know). Most recently, it
appeared in \cite{GonLakLadDetSchub} and~\cite{GoncMilMixLadDet},
although the constructions in these papers and \cite{MulayLadders} are
not exactly the same as procedure~\ref{goncmil}. Instead, their methods
involve constructing a parabolic subgroup $Q \subset \gln$ from the
vexillary essential set, and then associating a coset of the Weyl
group~$W_Q$ to the ladder determinantal ideal. Translated into our
language
% For complete clarity and ease of comparison, here is how to translate
% the notation in the above two references to agree with ours: if $Q^+$
% is a block upper-triangular parabolic, then their $\gln/Q^+$
% corresponds here to its transpose $Q\dom\gln$; their Schubert varieties
% $B_+wQ^+$ in $\gln/Q^+$ correspond to our Schubert varieties
% $Q(w_0w)^TB_+$ in $Q\dom\gln$; and their oppposite big cell $B_-w_0Q^+$
% in $\gln/Q^+$ corresponds to our opposite big cell $Qw_0^TB$ in
% $Q\dom\gln$.
(that is, using block {\em lower} triangular parabolic subgroups~$Q$
instead of their upper-triangular transposes, permutations $w_0w$ in
place of what they call~$w$, and partial flag varieties $Q\dom\gln$), the
preimage in $\gln$ of their constructed Schubert variety in $Q\dom\gln$
is $\wt X_w$, where $w$ is the shortest representative of its coset.
Intersecting this $\wt X_w$ with the opposite unipotent radicals of $B$
and $Q$ yields two varieties in different affine spaces whose ideals are
both extended from $\ol I_w$. Our main point from this perspective is
that one can always find an appropriate opposite big Schubert cell in
$B\dom\gln$, without passing to $Q\dom\gln$.
The original connection between determinantal ideals and geometry of
partial flag manifolds came from the Schubert subvarieties on Grassmann
varieties. Granted the many appearances of partitions in this context,
it seems natural in hindsight that Sturmfels was able to make his
fundamental application of the Knuth--Robinson--Schensted correspondence
to prove results on ideals generated by same-size determinants in
rectangular matrices \cite{StuGBdetRings}. It also seems natural from
this point of view that Herzog and Trung were able to extend Sturmfels'
methods to ideals cogenerated by fixed minors \cite{HerzogTrung}, because
these correspond to the open cells of Schubert varieties in
Grassmannians.
The application of KRS techniques to the algebra of determinantal rings
has since become quite an industry, to the point where alternative
methods seem somewhat scarce (see the Introduction). However, we found
ourselves unable to apply KRS techniques to the Schubert determinantal
ideals; perhaps others will succeed where we have failed. That being
said, note that the term orders driving the KRS methods differ in one
fundamental respect from ours: in our language, {\em diagonal\/} term
orders rather than antidiagonal term orders permeate the papers of
Sturmfels, Herzog--Trung, and others building on their work. As noted in
Example~\ref{ex:13254}, the minors in a determinantal ideal fail to be
Gr\"obner bases for diagonal term orders as soon as the rank conditions
become nested, whereas nested minors have relatively prime antidiagonal
terms. Since ladder determinantal ideals are precisely those whose rank
conditions are not nested (Proposition~\ref{prop:vex}), this leads us to
believe that diagonal term orders nearly approach their limits in
treatments such as \cite{GonLakLadDetSchub,GoncMilMixLadDet}.
On the other hand, our use of antidiagonal term orders does not allow us
to draw any new conclusions about ideals generated by determinants of
submatrices whose entries lie on a two-sided ladder. By definition, a
\bem{two-sided ladder} contains any square submatrix as soon as it
contains the main diagonal of the submatrix. However, we can use the
elimination methods of Herzog and Trung \cite{HerzogTrung} to prove
similar results for the analogous \bem{two-sided antiladders}, which
contain a square submatrix as soon as they contain the main antidiagonal.
We have left such generalizations open because this paper is already
quite lengthy, and because we have no obvious comments on the intrinsic
interest of such ideals. For instance, it seems unlikely that there is a
clear connection to the geometry of Schubert varieties, although their
initial ideals (for antidiagonal term orders, of course) might be
combinatorially interesting.
\subsection{Degeneracy loci}\label{sub:loci}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We recall here Fulton's theory of degeneracy loci, and explain its
relation to equivariant cohomology (see also the Appendix).
This was our initial interest in
this subject (and while not strictly an ``application of the Gr\"obner
basis,'' it does give another motivation for wanting formulae for
double Schubert polynomials).
Since completing this work, however, we learned of
the papers \cite{FRthomPoly,Kaz97} taking essentially the same viewpoint,
and we refer to them for detail.
Given a flagged vector bundle
$E_\spot = (E_1 \into E_2 \into \cdots \into E_n)$
and a co-flagged vector bundle
$F_\spot = (F_n \onto F_{n-1} \onto \cdots \onto F_1)$ over the same base $X$,
a generic map $\sigma : E_n \to F_n$, and a permutation $w$,
define the \bem{degeneracy locus} $\WW_w$ as the subset
\begin{eqnarray*}
\WW_w &=& \{x\in X \mid \rank (E_q \to E_n \stackrel \sigma \too F_n
\onto F_p) \leq \rank(w^T\sub qp) \hbox{ for all } q,p\}.
\end{eqnarray*}
The principal goal in Fulton's paper \cite{FulDegLoc} was to provide
``formulae for degeneracy loci'' as polynomials in the Chern classes
of the vector bundles.
In terms of the Chern roots
$\{ c_1(E_p/E_{p-1}), c_1(\ker F_q \to F_{q-1}) \}$,
Fulton found that the desired polynomials were actually the
double Schubert polynomials.
It is initially surprising that there is a single formula, for all
$X,E,F$ and not really depending on $\sigma$. This follows from a
classifying space argument, at least when $\kk = \CC$, as follows.
The group of automorphisms of a flagged vector space is the lower
triangulars $B$, and so the classifying space $BB$ of $B$-bundles carries
a universal flagged vector bundle. The classifying space of interest to
us is thus $BB \times BB_+$, which carries a pair of universal vector
bundles $\mathcal E$ and $\mathcal F$, the first flagged and the second
co-flagged. We write $\hhom({\mathcal E},{\mathcal F})$ for the bundle
whose fiber at $(x,y)\in BB\times BB_+$ is $\hhom({\mathcal
E}_x,{\mathcal F}_y)$.
Define the \bem{universal degeneracy locus} $U_w \subseteq
\hhom({\mathcal E},{\mathcal F})$ as the subset
\begin{eqnarray*}
U_w &=& \{(x,y,\phi) \mid \rank (({\mathcal E}_x)_q \stackrel \phi \too
({\mathcal F}_y)_p) \leq \rank(w^T\sub qp) \hbox{ for all } q,p\},
\end{eqnarray*}
where \hbox{$x \in BB$}, \hbox{$y \in BB_+$}, and $\phi : \EE_x \to
\FF_y$. In other words, the homomorphisms in the fiber of $U_w$ at
$(x,y)$ lie in the corresponding matrix Schubert variety.
The name is justified by the following. Recall that our setup is a space
$X$, a flagged vector bundle $E$ on it, a coflagged vector bundle $F$,
and a `generic' vector bundle map $\sigma : E \to F$; we will soon see
what `generic' means. Pick a classifying map $\chi : X \to BB \times
BB_+$, which means that $E,F$ are isomorphic to pullbacks of the
universal bundles. (Classifying maps exist uniquely up to homotopy.)
Over the target we have the universal $\Hom$-bundle $\hhom({\mathcal
E},{\mathcal F})$, and the vector bundle map $\sigma$ is a choice of a
way to factor the map $\chi$ through a map $\tilde\sigma : X \to
\hhom({\mathcal E},{\mathcal F})$. The degeneracy locus $\WW_w$ is then
$\tilde\sigma^{-1}(U_w)$, and it is natural to request that
$\tilde\sigma$ be transverse to each $U_w$---this will be the notion of
$\sigma$ being generic.
What does this say cohomologically? The closed subset $U_w$ defines a
class in Borel-Moore homology (and thus ordinary cohomology) of the
$\Hom$-bundle $\hhom({\mathcal E},{\mathcal F})$ (see Appendix
\ref{app:cohomology} for one proof of this). If $\sigma$ is generic,
then
\begin{eqnarray*}
[\WW_w] &=& [ \tilde\sigma^{-1}(U_w) ] \ \:=\ \:\tilde\sigma^*([U_w])
\end{eqnarray*}
and this is the sense in which there is a universal formula $[U_w] \in
H^*(\hhom({\mathcal E},{\mathcal F}))$. The cohomology ring of this
$\Hom$-bundle is the same as that of the base $BB \times BB_+$ (to which
it retracts), namely a polynomial ring in the $2n$ first Chern classes,
so one knows a priori that the universal formula should be expressible as
a polynomial in these $2n$ variables.
We can rephrase this using Borel's mixing space definition of
equivariant cohomology. Given a space $S$ carrying an action of a
group $G$, and a contractible space $EG$ upon which $G$ acts freely, define
the equivariant cohomology $H^*_G(S)$ of $S$ as
\begin{eqnarray*}
H^*_G(S) &:=& H^*( (S \times EG)/G)
\end{eqnarray*}
where the quotient is respect to the diagonal action. Note that the
`mixing space' $(S\times EG)/G$ is a bundle over $EG/G =: BG$, with
fibers $S$. In particular $H^*_G(S)$ is automatically a module over
$H^*(BG)$, thereby called the `base ring' of $G$-equivariant cohomology.
For us, the relevant group is $B\times B_+$, and we have two spaces $S$;
the space of matrices $\mn$ under left and right multiplication, and
inside it the matrix Schubert variety $\ol X_w$. Applying the mixing
construction to the pair $\mn \supseteq \ol X_w$, it can be shown that we
recover the bundles $\hhom({\mathcal E},{\mathcal F}) \supseteq U_w$. As
such, the universal formula $[U_w] \in H^*(\hhom({\mathcal E},{\mathcal
F}))$ we seek can be viewed instead as the class defined in $(B\times
B_+)$-equivariant cohomology by $\ol X_w$ inside $\mn$. As we prove in
Theorem~\ref{thm:positive} (in the setting of multidegrees; see
Theorem~\ref{notthm:oracle} for the direct equivariant cohomological
version), these are the double Schubert polynomials.
We point out the (few) differences between this approach and
that of Fulton in \cite{FulDegLoc}.
In the algebraic category, where Fulton worked, some pairs $(E,F)$ of
algebraic vector bundles may have no {\em algebraic} generic maps $\sigma$.
The derivation given above works more generally
in the topological category, where no restriction on $(E,F)$ is necessary.
Secondly, we don't even need to know, a priori, which polynomials
represent the cohomology classes of matrix Schubert varieties to show
that these classes are the universal degeneracy locus classes. This
contrasts with methods relying on divided differences.
%\end{section}{Applications of the Gr\"obner basis}%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Mitosis, rc-graphs, and subword complexes}%%%%%%%%%%%%%%%%%%%%%%
\label{sec:rc}
Sections~\ref{sub:pipe} and~\ref{sub:rc} translate some of the more
combinatorial parts of Section~\ref{sec:gb} into a language compatible
with rc-graphs (which don't fully appear until Definition~\ref{defn:rc}).
Section~\ref{sub:mitosis} uses the translation to produce an inductive
algorithm for generating rc-graphs, while Section~\ref{sub:combin}
provides an independent fully combinatorial proof. Section~\ref{sub:cm}
introduces a new family of simplicial complexes (subword complexes, in
Definition~\ref{defn:subword}), with the initial complexes $\LL_w$ as
special cases, and exploits their shellability to give a new proof of
Cohen--Macaulayness for Schubert varieties. After proving that subword
complexes are balls or spheres in Section~\ref{sub:balls}, the final
Section~\ref{sub:links} employs Stanley-Reisner theory for subword
complexes to elucidate the combinatorics of Grothendieck polynomials.
\subsection{Pipe dreams and mitosis}\label{sub:pipe}%%%%%%%%%%%%%%%%%%%%%
Consider a square grid $\ZZ_{> 0} \times \ZZ_{> 0}$ extending infinitely
south and east, with the box in row~$i$ and column~$j$ labeled $(i,j)$,
as in an $\infty \times \infty$ matrix. If each box in the grid is
covered with a square tile containing either $\textcross$ or
$\textelbow$, then one can think of the tiled grid as a network of pipes.
\begin{defn} \label{defn:pipe}
A \bem{pipe dream}%
%
\footnote{In the game Pipe Dream, the player is supposed to guide
water flowing out of a spigot at one edge of the game board to
its destination at another edge by laying down given square tiles
with pipes going through them.
% This also explains the term \bem{elbow joints}.
Definition~\ref{defn:rc} interprets the spigot placements and
destinations.}
%
is a finite subset of $\ZZ_{> 0} \times \ZZ_{> 0}$, identified as the set
of crosses in a tiling by \bem{crosses} and \bem{elbow joints}. Given an
$n \times n$ exponent array~$\bb$, let $D(\bb) = [n]^2 \minus
\supp(\zz^\bb)$ be the \bem{pipe dream associated to $\bb$}.%
\end{defn}
To readers familiar with rc-graphs, which we will recall in
Definition~\ref{defn:rc}, we point out that
not all pipe dreams are rc-graphs; for
instance, the $n=5$ pipe dreams in Example~\ref{ex:pipe} are not
rc-graphs. Unless otherwise stated, all pipe dreams are assumed to be
contained in the triangular region $\{(q,p) \in [n]^2 \mid q+p \leq n\}$
strictly above the main antidiagonal. Whenever we draw pipe dreams, we
fill the boxes with crossing tiles by \cross. However, we often leave
the elbow tiles blank, or denote them by dots for ease of notation.
\begin{example} \label{ex:pipe}
If $\zz^\bb = \prod_{q+p \leq n} z_{qp}$, then $D(\bb)$ is a pipe dream
called $D_0$, with crosses strictly above the main antidiagonal and elbow
joints elsewhere. Here are two rather arbitrary pipe dreams with $n=5$:
$$
\begin{array}{|c|c|c|c|c|}
\multicolumn{5}{c}{}
\\\hline\!+\!& &\!+\!&\!+\!&\ \,
\\\hline & & & &
\\\hline\!+\!&\!+\!& & &
\\\hline & & & &
\\\hline & & & &
\\\hline
\end{array}
\,\quad =
\begin{array}{cccccc}
&\phperm &\phperm &\phperm &\phperm &\phperm \\
& \+ & \jr & \+ & \+ & \je \\
& \jr & \jr & \jr & \je &\\
& \+ & \+ & \je & &\\
& \jr & \je & & &\\
& \je & & & &
\end{array}
\quad \hbox{and} \qquad
\begin{array}{|c|c|c|c|c|}
\multicolumn{5}{c}{}
\\\hline &\!+\!& &\!+\!&\ \,
\\\hline &\!+\!&\!+\!& &
\\\hline\!+\!& & & &
\\\hline\!+\!& & & &
\\\hline & & & &
\\\hline
\end{array}
\,\quad =
\begin{array}{cccccc}
&\phperm &\phperm &\phperm &\phperm &\phperm \\
& \jr & \+ & \jr & \+ & \je \\
& \jr & \+ & \+ & \je &\\
& \+ & \jr & \je & &\\
& \+ & \je & & &\\
& \je & & & &
\end{array}
$$
Another (slightly less arbitrary) example, with $n=8$, is $D =$
$$
\begin{array}{|c|c|c|c|c|c|c|c|}
\multicolumn{8}{c}{}
\\\hline &\!+\!& &\!+\!&\!+\!& &\ \, &\ \,
\\\hline &\!+\!& & & &\!+\!& &
\\\hline\!+\!&\!+\!&\!+\!&\!+\!& & & &
\\\hline & &\!+\!& & & & &
\\\hline\!+\!& & & & & & &
\\\hline\!+\!&\!+\!& & & & & &
\\\hline\!+\!& & & & & & &
\\\hline & & & & & & &
\\\hline
\end{array}
\ \quad = \quad
\begin{array}{ccccccccc}
&\perm w1&\perm w8&\perm w2&\perm w7&\perm w5&\perm w4&\perm w6&\perm w3\\
\petit1& \jr & \+ & \jr & \+ & \+ & \jr & \jr & \je\\
\petit2& \jr & \+ & \jr & \jr & \jr & \+ & \je &\\
\petit3& \+ & \+ & \+ & \+ & \jr & \je & &\\
\petit4& \jr & \jr & \+ & \jr & \je & & &\\
\petit5& \+ & \jr & \jr & \je & & & &\\
\petit6& \+ & \+ & \je & & & & &\\
\petit7& \+ & \je & & & & & &\\
\petit8& \je & & & & & & &
\end{array}
$$
in which the first diagram represents $D$ as a subset of $[8]^2$, whereas
the second demonstrates how the tiles fit together. Since no cross
in~$D$ occurs on or below the $8^\th$ antidiagonal, the pipe entering
row~$i$ exits column~$w_i = w(i)$ for some permutation $w \in S_8$. In
this case, $w = 13865742$. For clarity, we omit the square tile
boundaries as well as the ``sea'' of elbows below the main antidiagonal
in the right pipe dream. We also use the thinner symbol $w_i$ instead of
$w(i)$ to make the widths come out right.%
\end{example}
For the rest of this subsection we think of the row index~$i$ as being
fixed, just as we did in Sections~\ref{sub:mutation}--\ref{sub:Lw}.
Given a pipe dream in $[n]^2$, define the column index
\begin{eqnarray} \label{eq:pipestart}
\start_i(D) &=& \min(\{j \mid (i,j) \not\in D\} \cup \{n+1\})
\end{eqnarray}
of the westernmost empty box in row $i$ of $D \subseteq [n]^2$. The
similarity with the notation in Section~\ref{sub:lift} is explained by
the following lemma, whose proof is immediate from the definitions.
\begin{lemma} \label{lemma:max}
$\start_i(\bb) = \west_i(\bb) = \start_i(D(\bb))$ if $\zz^\bb \not\in
J_w$ has maximal support.
\end{lemma}
Most of the rest of Sections~\ref{sub:pipe}--\ref{sub:mitosis} concerns
the relation between mutation (Definition~\ref{defn:mu}) and
\bem{mitosis},%
%
\footnote{The term \bem{mitosis} is biological lingo for cell
division in multicellular organisms.}
%
which we now define.
\begin{defn} \label{defn:mitosis}
Given a pipe dream $D \subseteq [n]^2$ and a row index~$i$, construct a
set $\mitosis_i(D)$ of pipe dreams contained in $[n]^2$ as follows. Set
\begin{eqnarray*}
\JJ(D) &=& \{j < \start_i(D) \mid (i+1,j) \not\in D\};\\
\JJ_{i,\leq p} &=& \{(i,j) \mid j \in \JJ(D) \hbox{ and } j \leq p\}
\hbox{ for } p \in \JJ(D);\\
\JJ_{i+1,< p} &=& \{(i+1,j) \mid j \in \JJ(D) \hbox{ and } j < p\} \hbox{
for } p \in \JJ(D).
\end{eqnarray*}
Define the \bem{offspring} $D_p = D \cup \JJ_{i+1,< p} \minus \JJ_{i,\leq
p}$ and $\mitosis_i(D) = \{D_p \mid p \in \JJ(D)\}$. Given a set
${\mathcal P}$ of pipe dreams, write $\mitosis_i({\mathcal P}) =
\bigcup_{D \in {\mathcal P}} \mitosis_i({\mathcal P})$.%
\end{defn}
For a pipe dream $D$, the offspring $D_p$ is obtained by first deleting
the cross at $(i,p)$ in $D$, and then moving the crosses west of it from
row~$i$ south to empty boxes in row~$i+1$ (mitosis only cares about the
columns $\JJ(D)$, which are empty in row~$i+1$). In the region of $D$
that is west of $(i,p)$, row~$i$ is filled solidly with crosses. Observe
that $\mitosis_i(D)$ is an empty set whenever $\JJ(D)$ is empty.
Proposition~\ref{prop:offspring} presents another verbal description of
mitosis---this one a little more algorithmic---using a certain local
transformation on pipe dreams that was discovered by Bergeron and Billey.
\begin{defnlabeled}{\cite{BB}} \label{defn:chute}
A \bem{chutable rectangle} is a connected $2 \times k$ rectangle $C$
inside a pipe dream $D$ such that $k \geq 2$ and all but the following 3
locations in $C$ are crosses: the northwest, southwest, and southeast
corners. Applying a \bem{chute move}%
%
\footnote{The transpose of a chute move is called a \bem{ladder
move} in \cite{BB}.}
%
to $D$ is accomplished by placing a \cross in the southwest corner of a
chutable rectangle $C$ and removing the \cross from the northeast corner of
the same~$C$.%
\end{defnlabeled}
Heuristically, a chute move therefore looks like:
\begin{rcgraph}
% 1 2 3 4 5 6 7 8 9
\begin{array}{@{}r|c|c|c|@{}c@{}|c|c|c|l@{}}
\multicolumn{7}{c}{}&\multicolumn{1}{c}{
\phantom{\!+\!}}&
\multicolumn{1}{c}{\begin{array}{@{}c@{}}\\\adots\end{array}}
\\\cline{2-8}
&\cdot&\!+\!&\!+\!& \toplinedots &\!+\!&\!+\!&\!+\!
\\\cline{2-4}\cline{6-8}
&\cdot&\!+\!&\!+\!& &\!+\!&\!+\!&\cdot
\\\cline{2-8}
\multicolumn{1}{c}{\begin{array}{@{}c@{}}\adots\\ \\ \end{array}}&
\multicolumn{1}{c}{\phantom{\!+\!}}
\end{array}
%
\quad\stackrel{\rm chute}\rightsquigarrow\quad
%
% 1 2 3 4 5 6 7 8 9
\begin{array}{@{}r|c|c|c|@{}c@{}|c|c|c|l@{}}
\multicolumn{7}{c}{}&\multicolumn{1}{c}{
\phantom{\!+\!}}&
\multicolumn{1}{c}{\begin{array}{@{}c@{}}\\\adots\end{array}}
\\\cline{2-8}
&\cdot&\!+\!&\!+\!& \toplinedots &\!+\!&\!+\!&\cdot
\\\cline{2-4}\cline{6-8}
&\!+\!&\!+\!&\!+\!& &\!+\!&\!+\!&\cdot
\\\cline{2-8}
\multicolumn{1}{c}{\begin{array}{@{}c@{}}\adots\\ \\ \end{array}}&
\multicolumn{1}{c}{\phantom{\!+\!}}
\end{array}
\end{rcgraph}
\begin{prop} \label{prop:offspring}
Let $D$ be a pipe dream, and suppose $j$ is the smallest column index
such that $(i+1,j) \not\in D$ and $(i,p) \in D$ for all $p \leq j$. Then
$D_p \in \mitosis_i(D)$ is obtained from $D$ by
\begin{thmlist}
\item
removing $(i,j)$, and then
\item
performing chute moves from row~$i$ to row~$i+1$, each one as far west as
possible, so that $(i,p)$ is the last {\rm \cross$\!$} removed.
\end{thmlist}
\end{prop}
\begin{proof}
Immediate from Definitions~\ref{defn:chute} and~\ref{defn:mitosis}.
\end{proof}
\begin{example} \label{ex:mitosis}
Fig.~\ref{fig:mitosis} depicts the algorithmic `chute' form of mitosis in
Proposition~\ref{prop:offspring}, with $i = 3$. The dot in each pipe
dream represents the last \cross removed.%
\begin{figure}[t]
\begin{rcgraph}
\begin{array}{ccc}
%
\begin{array}{@{}cc@{}}{}\\\\3&\\4\\\\\\\\\\\end{array}
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline &\!+\!& &\!+\!&\!+\!& &\ \, &\ \,
\\\hline &\!+\!& & & &\!+\!& &
\\\hline\!+\!&\!+\!&\!+\!&\!+\!& & & &
\\\hline & &\!+\!& & & & &
\\\hline\!+\!& & & & & & &
\\\hline\!+\!&\!+\!& & & & & &
\\\hline\!+\!& & & & & & &
\\\hline & & & & & & &
\\\hline
\end{array}
%
&\longmapsto&
%
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline &\!+\!& &\!+\!&\!+\!& &\ \, &\ \,
\\\hline &\!+\!& & & &\!+\!& &
\\\hline\cdot&\!+\!&\!+\!&\!+\!& & & &
\\\hline & &\!+\!& & & & &
\\\hline\!+\!& & & & & & &
\\\hline\!+\!&\!+\!& & & & & &
\\\hline\!+\!& & & & & & &
\\\hline & & & & & & &
\\\hline
\end{array}
%
\\\\[-8pt]
%
\quad\ D
%
& &
%
\downarrow
%
\\[4pt]
%
& &
%
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline &\!+\!& &\!+\!&\!+\!& &\ \, &\ \,
\\\hline &\!+\!& & & &\!+\!& &
\\\hline &\cdot&\!+\!&\!+\!& & & &
\\\hline\!+\!& &\!+\!& & & & &
\\\hline\!+\!& & & & & & &
\\\hline\!+\!&\!+\!& & & & & &
\\\hline\!+\!& & & & & & &
\\\hline & & & & & & &
\\\hline
\end{array}
%
\\\\[-8pt]
%
& &
%
\downarrow
%
\\[4pt]
%
& &
%
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline &\!+\!& &\!+\!&\!+\!& &\ \, &\ \,
\\\hline &\!+\!& & & &\!+\!& &
\\\hline & &\!+\!&\cdot& & & &
\\\hline\!+\!&\!+\!&\!+\!& & & & &
\\\hline\!+\!& & & & & & &
\\\hline\!+\!&\!+\!& & & & & &
\\\hline\!+\!& & & & & & &
\\\hline & & & & & & &
\\\hline
\end{array}
\end{array}
\end{rcgraph}
\caption{Mitosis} \label{fig:mitosis}
\end{figure}%
\end{example}
Proposition~\ref{prop:offspring} for mitosis has the following analogue
for mutation.
\begin{claim} \label{claim:mu}
Suppose that $\length(ws_i) < \length(w)$, and let $\zz^\bb \not\in J_w$
be a squarefree monomial of maximal support. If $D = D(\bb)$ then
\begin{eqnarray*}
|\prom(\bb)| &=& |\JJ(D)|.
\end{eqnarray*}
If $0 \leq d < |\prom(\bb)|$, then $D(\mu_i^{2d+1}(2\bb))$ is obtained
from $D$ by
\begin{thmlist}
\item
removing $(i,j)$, where $j$ is as in Proposition~\ref{prop:offspring},
and then
\item
performing $d$ chute moves from row~$i$ to row~$i+1$, each as far west as
possible.
\end{thmlist}
\end{claim}
\begin{proof}
By Lemma~\ref{lemma:max}, the columns in $\JJ(D)$ are in bijection with
the nonzero entries in the promoter of $\bb$, each of which is a~$1$ in
row~$i+1$. The final statement follows easily from the definitions.
\end{proof}
\begin{example} \label{ex:mitosis'}
The array $\bb$ in Example~\ref{ex:mutate} has maximal support among
exponent arrays on monomials not in $J_w$, where $w = 13865742$ as in
Examples~\ref{ex:prom} and~\ref{ex:intro}. Substituting \cross for each
blank space and then removing the numbers and dots yields the pipe dream
$D = D(\bb)$ in Fig.~\ref{fig:mitosis}. Applying the same makeover to
the middle column of Fig.~\ref{fig:mu} results in the offspring
$D(\mu_3^1(\bb))$, $D(\mu_3^3(\bb))$, and $D(\mu_3^5(\bb))$.%
\end{example}
\begin{lemma} \label{lemma:agree}
If $\length(ws_i) < \length(w)$ and $\zz^\bb \not\in J_w$ is squarefree
of maximal support, then $\mitosis_i(D(\bb)) = \{D(\mu_i^{2d+1}(2\bb))
\mid 0 \leq d < |\prom(\bb)|\}$.
\end{lemma}
\begin{proof}
Compare Claim~\ref{claim:mu} with Proposition~\ref{prop:offspring}.
% Observe that both produce the same number of new pipe dreams: $|\JJ(D)|
% = |\prom(\bb)|$.
\end{proof}
\subsection{Facets of $\LL_w$ and rc-graphs}\label{sub:rc}%%%%%%%%%%%%%%%
Proposition~\ref{prop:mitosis}, in which $D_L = [n]^2 \minus L$ for $L
\in \LL_w$ as per the beginning of Section~\ref{sub:Lw}, will conclude
the translation of mutation on monomials into mitosis on facets. Its
full potential will be realized in Theorem~\ref{thm:rc} and especially
the algorithm in Theorem~\ref{thm:mitosis}. The translation requires an
intermediate result.
\begin{lemma} \label{lemma:facets}
If $\length(ws_i) < \length(w)$, then $\{D_L \mid L$ is a facet of
$\LL_{ws_i}\}$ is the set of pipe dreams $D(\mu_i^{2d+1}(2\bb))$ such
that $\zz^\bb \not\in J_w$ is squarefree of maximal support and $0 \leq d
< |\prom(\bb)|$.
\end{lemma}
\begin{proof}
By Proposition~\ref{prop:ev} and Lemma~\ref{lemma:notafacet}, every facet
of $\LL_{ws_i}$ is the support of a mutation $\mu_i^d(\zz^{2\bb})$ for
some monomial $\zz^{2\bb} \not\in J_w$ and $d \leq |\prom(2\bb)|$.
Furthermore, it is clear from Lemma~\ref{lemma:notafacet} and the
definition of mutation that we may assume $\zz^\bb$ is a squarefree
monomial (so the entries of $2\bb$ are all~$0$ or~$2$). In this case,
the supports of odd mutations $\mu_i^{2d+1}(\zz^{2\bb})$ for $0 \leq d <
|\prom(\bb)|$ are facets of $\LL_{ws_i}$ by Proposition~\ref{prop:pure}
because they each have cardinality $n^2 - \length(w) + 1 = n^2 -
\length(ws_i)$, while the supports of even mutations
$\mu_i^{2d}(\zz^{2\bb})$ for $d \leq |\prom(\bb)|$ aren't facets, each
having cardinality $n^2 - \length(w)$.
\end{proof}
\begin{prop} \label{prop:mitosis}
If $\length(ws_i) < \length(w)$, then
\begin{eqnarray*}
\{D_L \mid L \hbox{ is a facet of } \LL_{ws_i}\} &=& \mitosis_i(\{D_L
\mid L \hbox{ is a facet of } \LL_w\}).
\end{eqnarray*}
Moreover, $\mitosis_i(D_L) \cap \mitosis_i(D_{L'}) = \nothing$ if $L \neq
L'$ are facets of $\LL_w$.
\end{prop}
\begin{proof}
The displayed equation is a consequence of Lemma~\ref{lemma:facets} and
Lemma~\ref{lemma:agree}, so we concentrate on the final statement. Let
$\zz^\bb \not\in J_w$ be a squarefree monomial of maximal support $L =
\supp(\bb)$, and let $0 \leq d < |\prom(\bb)|$. The entries of the array
$\mu_i^{2d+1}(2\bb)$ are all either~$0$ or~$2$, except for precisely two
$1$'s, both in the same column~$p$ (the boldface entries in
Fig.~\ref{fig:mu}, middle column). By Lemma~\ref{lemma:max},~$p$ is the
westernmost column of $D(\mu_i^{2d+1}(2\bb))$ in which neither row~$i$
nor row~$i+1$ has a cross.
Now suppose $\zz^{\bb'} \not\in J_w$ is another squarefree monomial of
maximal support $L'$, and let $0 \leq d' < |\prom(\bb')|$. If
$D(\mu_i^{2d+1}(2\bb)) = D(\mu_i^{2d'+1}(2\bb'))$, then the argument in
the first paragraph of the proof implies that $\mu_i^{2d+1}(2\bb) =
\mu_i^{2d'+1}(2\bb')$, since they have the same entries equal to $1$ as
well as the same support, and all of their other nonzero entries
equal~$2$. We conclude that $\bb = \bb'$ by Lemma~\ref{lemma:unique}.
Using Lemma~\ref{lemma:facets}, we have proved that $\mitosis_i(D_L) \cap
\mitosis_i(D_{L'}) \neq \nothing$ implies $L = L'$.
\end{proof}
\begin{example} \label{ex:remove}
The pipe dream $D$ in Example~\ref{ex:mitosis'} and
Fig.~\ref{fig:mitosis} is $D_L$ for a facet of $\LL_{13865742}$. By
Proposition~\ref{prop:mitosis}, the three pipe dreams in the right column
of Fig.~\ref{fig:mitosis} can be expressed as $D_{L'}$ for facets $L' \in
\LL_{13685742}$, where $13685742 = 13865742 \cdot s_3$.%
\end{example}
Next is a result whose proof connects chuting with antidiagonals. It
will be used in the proof of Theorem~\ref{thm:rc}.
\begin{lemma} \label{lemma:chuting}
The set $\{D_L \mid L \in \facets(\LL_w)\}$ is closed under chute moves.
\end{lemma}
\begin{proof}
A pipe dream $D$ is equal to $D_L$ for some (not necessarily maximal) $L
\in \LL_w$ if and only if $D$ meets every antidiagonal in $J_w$, which by
definition of $\LL_w$ equals $\bigcap_{L \in \LL_w} \$. Supposing that $C$ is a chutable rectangle in $D_L$ for $L
\in \LL_w$, it is therefore enough to show that the intersection $a \cap
D_L$ of any antidiagonal $a \in J_w$ with $D_L$ does not consist entirely
of the single cross in the northeast corner of $C$. Indeed, the purity
of $\LL_w$ (Proposition~\ref{prop:pure}) will then imply that chuting
$D_L$ in $C$ yields $D_{L'}$ for some \bem{facet} $L'$ whenever $L \in
\facets(\LL_w)$.
To prove the claim concerning $a \cap D_L$, we may assume $a$ contains
the cross in the northeast corner $(q,p)$ of $C$, and split into cases:
\begin{textlist}
\item
$a$ does not continue south of row~$q$.
\item
$a$ continues south of row~$q$ but skips row~$q+1$.
\item
$a$ intersects row~$q+1$, but strictly east of the southwest corner of
$C$.
\item
$a$ intersects row~$q+1$ at or west of the southwest corner of $C$.
\end{textlist}
Letting $(q+1,t)$ be the southwest corner of $C$, construct new
antidiagonals $a'$ that are in $J_w$ (and hence intersect $D_L$) by
replacing the cross at $(q,p)$ with a cross at:
\begin{textlist}
\item
$(q,t)$, using Lemma~\ref{lemma:squish}(W);
\item
$(q+1,p)$, using Lemma~\ref{lemma:squish}(S);
\item
$(q,p)$, so $a = a'$ trivially; or
\item
$(q+1,t)$, using Lemma~\ref{lemma:squish}(E).
\end{textlist}
Observe that in case~(iii), $a$ already shares a spot in row~$q+1$ where
$D_L$ has a cross. Each of the other antidiagonals $a'$ intersects both
$a$ and $D_L$ in some spot that isn't $(q,p)$, since the location of $a'
\minus a$ has been constructed not to be a cross in $D_L$.%
\end{proof}
\begin{defnlabeled}{\cite{FKyangBax}} \label{defn:rc}
An \bem{rc-graph} is a pipe dream in which each pair of pipes crosses at
most once. If $D$ is an rc-graph and $w \in S_n$ is the permutation such
that the pipe entering row~$i$ exits from column $w(i)$, then $D$ is said
to be an \bem{rc-graph for $w$}. The set of rc-graphs for~$w$ is denoted
by $\rc(w)$.
\end{defnlabeled}
The name `rc-graph' was coined by Bergeron and Billey \cite{BB}, although
Fomin and Kirillov introduced objects represented by rc-graphs (rotated
by $135^\circ$) in special cases. The ``rc'' stands for
``reduced-compatible'', and is justified using our next lemma. Given a
pipe dream $D$, say that a \cross at $(q,p) \in D$ sits on the
\bem{$i^\th$ antidiagonal} if $q+p-1 = i$. Let $Q(D)$ be the ordered
list of simple reflections $s_i$ corresponding to the antidiagonals on
which the crosses sit, starting from the northeast corner of $D$ and
reading {\em right to left} in each row, snaking down to the southwest
corner.
\begin{lemma} \label{lemma:subword}
If $D$ is a pipe dream, then multiplying the reflections in $Q(D)$ yields
the permutation $w$ such that the pipe entering row~$i$ exits
column~$w(i)$. Furthermore, $|D| \geq \length(w)$, with equality if and
only if $D \in \rc(w)$.
\end{lemma}
\begin{proof}
For the first statement, use induction on the number of crosses: adding a
\cross in the $i^\th$ antidiagonal at the end of the list switches the
destinations of the pipes beginning in rows~$i$ and~$i+1$. Each
inversion in~$w$ contributes at least one crossing in $D$, whence $|D|
\geq \length(w)$. The expression $Q(D)$~is reduced when $D$~is an
rc-graph because each inversion in~$w$ contributes $\leq 1$~crossings
in~$D$.
\end{proof}
Thus $Q(D)$ gives a {\em reduced} expression for~$w$ if and only if $D
\in \rc(w)$. The ordered list of row indices for the crosses in~$D$
(taken in the same order as before) is called a
``compatible sequence'' for the expression~$Q(D)$ (we won't need this
concept from \cite{BJS}).
\begin{example} \label{ex:rc}
The upper-left triangular pipe dream $D_0 \subset [n]^2$, with crosses
strictly above the main antidiagonal and elbows elsewhere, is in (and in
fact equals)~$\rc(w_0)$. The $8 \times 8$ pipe dream~$D$ in
Example~\ref{ex:pipe} is in~$\rc(13865742)$; this is the same pipe
dream~$D$ appearing in Examples~\ref{ex:mitosis'} and~\ref{ex:remove}.
We will show in Theorem~\ref{thm:rc} that the three pipe dreams in the
right hand column of Fig.~\ref{fig:mitosis} are in fact rc-graphs for
$13685742 = 13865742 \cdot s_3$.
In general, notice how pipe dreams whose crosses lie strictly above the
main antidiagonal in~$[n]^2$ are naturally {\em subwords} of~$Q(D_0)$,
while rc-graphs are naturally {\em reduced} subwords. This point of view
will take center stage in Section~\ref{sub:cm}.%
\end{example}
Lemma~\ref{lemma:subword} implies the following criterion for when
removing a~\cross from a pipe dream $D \in \rc(w)$ yields a pipe dream
in~$\rc(ws_i)$. Specifically, it concerns the removal of a cross at
$(i,j)$ from configurations that look like
$$
\begin{array}{lccccccccc}
&\perm 1{}&\perm{}{}&\perm{}{}&\perm{}{}&
\perm{\hskip-11.6pt \textstyle \cdots}{}&\perm{}{}&\perm{}{}&\perm j\
\\[3pt]
\petit{i} &\+ & \+ & \+ & \+ & \+ & \+ & \+ & \+ &\toplinedots
\\[-1pt]
\petit{i+1}&\+ & \+ & \+ & \+ & \+ & \+ & \+ & \jr
\end{array}
%
\begin{array}{c}\\ = \\\end{array}\
%
\begin{array}{l|c|c|c|c|c|c|c|c|c}
\multicolumn{1}{c}{}
&\multicolumn{1}{c}{\petit 1}
&\multicolumn{6}{c}{\cdots}
&\multicolumn{1}{c}{\petit j}
\\[3pt]\cline{2-10}
\petit{i} &\!+\!&\!+\!&\!+\!&\!+\!&\!+\!&\!+\!&\!+\!&\!+\!&\toplinedots
\\\cline{2-9}
\petit{i+1}&\!+\!&\!+\!&\!+\!&\!+\!&\!+\!&\!+\!&\!+\!&\cdot&\\\cline{2-10}
\end{array}
$$
at the west end of rows~$i$ and~$i+1$ in~$D$.
\begin{lemma} \label{lemma:rc}
Let $D \in \rc(w)$ and $j$ be a fixed column index with $(i+1,j) \not\in
D$, but $(i,p) \in D$ for all $p \leq j$, and $(i+1,p) \in D$ for all $p
< j$. Then $\length(ws_i) < \length(w)$, and if $D' = D \minus (i,j)$
then $D' \in \rc(ws_i)$.
\end{lemma}
\begin{proof}
Removing $(i,j)$ only switches the exit points of the two pipes starting
in rows $i$ and $i+1$, so the pipe starting in row $k$ of $D'$ exits out
of column $ws_i(k)$ for each $k$. The result follows from
Lemma~\ref{lemma:subword}.
\end{proof}
The connection beween the complexes $\LL_w$ and rc-graphs requires
certain facts proved by Bergeron and Billey \cite{BB}. The next lemma
consists mostly of the combinatorial parts (a),~(b), and~(c) of
\cite[Theorem~3.7]{BB}, their main result. Its proof relies exclusively
on elementary properties of rc-graphs.
\begin{lemma}[\cite{BB}] \label{lemma:BB} \mbox{}
\begin{thmlist}
\item \label{closed}
The set $\rc(w)$ of rc-graphs for~$w$ is closed under chute operations.
\item \label{unique}
There is a unique \bem{top rc-graph} for $w$ such that every cross not
in the first row has a cross due north of it.
\item
Every rc-graph for $w$ can be obtained by applying a sequence of chute
moves to the top rc-graph for $w$.
\end{thmlist}
\end{lemma}
\begin{lemma} \label{lemma:top}
The top rc-graph for~$w$ is $D_L$ for a facet $L \in \LL_w$.
\end{lemma}
\begin{proof}
The unique rc-graph for $w_0$, whose crosses are at $\{(q,p) \mid q+p
\leq n\}$, is also $D_L$ for the unique facet of $\LL_{w_0}$. Therefore,
if $\length(ws_i) < \length(w)$ we employ downward induction to prove the
result for $ws_i$ by assuming it for $w$. Using Lemma~\ref{lemma:lex}
(particularly part~\ref{zeros}) and Lemma~\ref{lemma:facet}, we assume
the top rc-graph $D \in \rc(w)$ satisfies the conditions of
Lemma~\ref{lemma:rc}, with $j = w(i+1)$. By definition, the column
index~$j$ is the minimal one in $\JJ(D)$. Thus $D \minus (i,j)$ equals
the first offspring $D_j$ from Definition~\ref{defn:mitosis} (or
Proposition~\ref{prop:offspring}) as well as the rc-graph $D' \in
\rc(ws_i)$ from Lemma~\ref{lemma:rc}. That $D'$ is the {\em top}
rc-graph for $ws_i$ uses part~\ref{unique} of Lemma~\ref{lemma:BB}, while
$D_j = D_L$ for some facet $L \in \LL_{ws_i}$ by
Proposition~\ref{prop:mitosis}.
\end{proof}
We come now to our first main theorem of Section~\ref{sec:rc}. It
ascribes a truly geometric origin to rc-graphs, identifying them with the
subspaces of $\mn$ resulting from flat deformation of matrix Schubert
varieties (Section~\ref{sec:gb}).
\begin{thm} \label{thm:rc}
$\rc(w) = \{D_L \mid L$ is a facet of $\LL_w\}$, where $D_L = [n]^2
\minus L$. In other words, rc-graphs for~$w$ are complements of maximal
supports of monomials $\not\in J_w$.
\end{thm}
\begin{proof}
Lemmas~\ref{lemma:top} and~\ref{lemma:chuting} imply that $\rc(w)
\subseteq \{D_L \mid L \in \facets(\LL_w)\}$, given Lemma~\ref{lemma:BB}.
Since the opposite containment $\{D_L \mid L \in \facets(\LL_v)\}
\subseteq \rc(v)$ is obvious for $v = w_0$, it suffices to prove it for
$v = ws_i$ by assuming it for $v = w$.
A pair $(i+1,j)$ as in Lemma~\ref{lemma:rc} exists in $D$ if and only if
the set $\JJ(D)$ in Definition~\ref{defn:mitosis} nonempty, and this
occurs if and only if $\mitosis_i(D) \neq \nothing$. In this case, the
pipe dream produced from $D$ by the first stage of $\mitosis_i$ (as in
Proposition~\ref{prop:offspring}) is $D' = D \minus (i,j) \in \rc(ws_i)$.
% , since $(i,j)$ is the westernmost \cross in $\JJ_{i,\leq s}$.
The desired containment follows from Proposition~\ref{prop:mitosis} and
part~\ref{closed} of Lemma~\ref{lemma:BB}.
\end{proof}
The next formula was our motivation for relating $\LL_w$ to~$\rc(w)$. It
was originally proved by Billey, Jockusch, and Stanley, but Fomin and
Stanley shortly thereafter gave a better combinatorial proof.
\begin{cor}[\cite{BJS,FSnilCoxeter}] \label{cor:BJS}
$\displaystyle \SS_w(\xx)\ = \sum_{D \in \rc(w)} \xx^D$.
\end{cor}
\begin{proof}
Theorem~\ref{thm:positive} expresses $\SS_w(\xx)$ as a sum of monomials
$[L]_T = \xx^{D_L}$ for $D_L$ in $\{[n]^2 \minus L \mid L$ is a facet of
$\LL_w\}$. Theorem~\ref{thm:rc} identifies this set as $\rc(w)$.
\end{proof}
Here is the double version of the previous result; its proof is the same.
\begin{cor}[\cite{FKyangBax}] \label{cor:doubSchub}
$\displaystyle \SS_w(\xx,\yy)\ = \sum_{D \in \rc(w)} \prod_{(q,p) \in D}
(x_q - y_p)$.
\end{cor}
\subsection{The mitosis algorithm for rc-graphs}\label{sub:mitosis}%%%%%%
The lifted Demazure operators from Section~\ref{sub:lift}, in their
mitosis avatar, produce an algorithm for obtaining coefficients on
monomials in Schubert polynomials by induction on the weak Bruhat order.
This already follows from Proposition~\ref{prop:mitosis} and
Theorem~\ref{thm:positive}. However, we express the algorithm in
Theorem~\ref{thm:mitosis} in terms of rc-graphs---which represent
$\ZZ^{n^2}$-graded data that is more refined than $\ZZ^n$-graded
coefficients---by way of Theorem~\ref{thm:rc}. The Schubert polynomial
interpretation of Theorem~\ref{thm:mitosis} via Corollary~\ref{cor:BJS}
serves as a geometrically motivated replacement for a conjecture of
Kohnert \cite[Appendix to Chapter IV, by N.\ Bergeron]{NoteSchubPoly}
concerning an algorithm for Schubert coefficients that is similarly
inductive, but on Rothe diagrams (Section~\ref{sub:other}) instead of
rc-graphs.
\begin{thm} \label{thm:mitosis}
$\rc(ws_i)$ is the disjoint union $\bigcupdot_{D \in \rc(w)}
\mitosis_i(D)$. Therefore, if $s_{i_1} \cdots s_{i_k}$ is a reduced
expression for $w_0w$ and $D_0$ is the unique rc-graph for $w_0$, then
\begin{eqnarray} \label{eq:alg}
\rc(w) &=& \mitosis_{i_k} \cdots \mitosis_{i_1}(D_0).
\end{eqnarray}
\end{thm}
\begin{proof}
Theorem~\ref{thm:rc} and Proposition~\ref{prop:mitosis}.
\end{proof}
\begin{remark} \label{rk:alg}
Bergeron and Billey used Lemma~\ref{lemma:BB} and Corollary~\ref{cor:BJS}
to give an algorithm for generating rc-graphs using chute moves
\cite[Theorem~3.7(d)]{BB}. Though motivated by and related to Kohnert's
conjecture, their algorithm is not inductive on the weak Bruhat order,
instead remaining inside $\rc(w)$ by starting from the top rc-graph
for~$w$ (Lemma~\ref{lemma:BB}). Some rc-graphs are produced twice in the
process.%
\end{remark}
The algorithm in Theorem~\ref{thm:mitosis} for generating $\rc(w)$ is
irredundant, in the sense that each rc-graph appears exactly once in the
implicit union on the right hand side of~(\ref{eq:alg}). However, the
efficiency of the algorithm can vary widely with the reduced expression
$s_{i_1} \cdots s_{i_k}$ for $w_0w$. For instance, the set $\rc(\id_n)$
for the identity permutation consists of one element, $\nothing \subset
[n]^2$, even though the repeated mitosis in~(\ref{eq:alg}) for $w =
\id_n$ can pass through $\rc(w)$ for {\em any} permutation $w$. As a
consequence, huge numbers of rc-graphs must be killed along the way,
without producing any offspring.
\begin{defn} \label{defn:poptotic}
A reduced expression for $w_0w$ is called \bem{poptotic}%
%
\footnote{The word `apoptosis' is a biological term referring
to ``programmed cell death'', in which some cell in a
multicellular organism in effect commits suicide for the greater
good of the organism. Thus \bem{apoptotic} is an apt term for a
total order in which some rc-graph dies along the way. In
analogy with the way `crepant' is formed from `discrepancy', we
take the opposite of `apoptotic' to be `poptotic'.}
%
if every rc-graph along the way to $\rc(w)$ has at least one offspring
via Theorem~\ref{thm:mitosis}. More precisely, the reduced expression
$w_0w = s_{i_1} \cdots s_{i_k}$ is poptotic if $\mitosis_{i_\ell}(D)$ is
nonempty whenever $1 < \ell \leq k$ and $D \in \rc(w_0 s_{i_1} \cdots
s_{i_{\ell-1}})$.
\end{defn}
\begin{example} \label{ex:apoptosis}
The three pipe dreams from the second column of Fig.~\ref{fig:mitosis}
are all rc-graphs for $v = 13685742$ (see Example~\ref{ex:rc}). Setting
$i = 4$ and either inspecting the inversions of $v$ or applying
Lemma~\ref{lemma:rc} to the last rc-graph in Fig.~\ref{fig:mitosis}, we
find that $\length(vs_4) < \length(v)$. On the other hand, $\mitosis_4$
kills the first two of the three rc-graphs, whereas the last has two
offspring. Thus any reduced expression for $w_0v$ ending $s_4$ is
necessarily apoptotic.
On the other hand, the lex first reduced expression of
Example~\ref{ex:lex}, which is $w_0v =
s_2s_1s_3s_5s_4s_3s_2s_1s_7s_6s_5s_4s_3s_2s_1$ in the present case, is
always poptotic by Lemmas~\ref{lemma:lex} and~\ref{lemma:rc}. In
particular, the lex first reduced expression for $w_0 = w_0 \id_n$ passes
through permutations with exactly one rc-graph (each is a \bem{dominant}
permutation, whose unqiue rc-graph is shaped like a Young diagram).%
\end{example}
Theorem~\ref{thm:mitosis} makes the set $\rc_n = \bigcup_{w \in S_n}
\rc(w)$ of rc-graphs for permutations of $n$ elements into a poset
determined by
\begin{eqnarray*}
D' \prec D &\hbox{ if }& D' \in \mitosis_i(D) \hbox{ for some } i.
\end{eqnarray*}
If this condition holds and $D \in \rc(w)$, then automatically $D' \in
\rc(ws_i)$, where $\length(ws_i) < \length(w)$ by Lemma~\ref{lemma:rc}.
Therefore the poset $\rc_n$, which is ranked by length = cardinality,
fibers over the weak Bruhat order on $S_n$, with the preimage of $w \in
S_n$ being $\rc(w)$.
A reduced decomposition for $w_0w$ can be thought of as a decreasing path
in the weak Bruhat order on $S_n$, beginning at $w_0$ and ending at $w$.
The preimage in $\rc_n$ of such a path is a tree having $\rc(w)$ among
its leaves. The path is poptotic, as in Definition~\ref{defn:poptotic},
when the leaves are precisely $\rc(w)$.
\begin{example} \label{ex:rc3}
Here is the Hasse diagram for $\rc_3$:
$$
\begin{array}{@{}r@{}c@{}l@{}}
\\[-4ex]
&
\hbox{\footnotesize 321}
\\[2pt]
&
\begin{tinyrc}{
\begin{array}{@{}|@{\,}c@{\,}|@{\,}c@{\,}|@{\,}c@{\,}|@{}}
\hline + & + &\phantom{+}
\\\hline + & &
\\\hline & &
\\\hline
\end{array}
}\end{tinyrc}
&
\\
\reflection{5pt} s2\diagup\,
&
&
\,\diagdown \reflection{5pt} s1
\\
\begin{array}{@{}c@{}}
\makebox[0pt][r]{\hbox{\footnotesize 312\ \ }}
\begin{tinyrc}{
\begin{array}{@{}|@{\,}c@{\,}|@{\,}c@{\,}|@{\,}c@{\,}|@{}}
\hline + & + &\phantom{+}
\\\hline & &
\\\hline & &
\\\hline
\end{array}
}\end{tinyrc}
\\
\reflection{2pt} s1/ \ \: \dom \reflection{2pt} s1
\\
\makebox[0pt][r]{\hbox{\footnotesize 132\ \ }}
\begin{tinyrc}{
\begin{array}{|@{\,}c@{\,}|@{\,}c@{\,}|@{\,}c@{\,}|}
\hline\phantom{+}& + &\phantom{+}
\\\hline & &
\\\hline & &
\\\hline
\end{array}
}\end{tinyrc}
\:
\begin{tinyrc}{
\begin{array}{|@{\,}c@{\,}|@{\,}c@{\,}|@{\,}c@{\,}|}
\hline &\phantom{+}&\phantom{+}
\\\hline + & &
\\\hline & &
\\\hline
\end{array}
}\end{tinyrc}
\end{array}
&
&
\phantom{\,\diagdown}
\begin{array}{@{\,}c@{}}
\begin{tinyrc}{
\begin{array}{@{}|@{\,}c@{\,}|@{\,}c@{\,}|@{\,}c@{\,}|@{}}
\hline + &\phantom{+}&\phantom{+}
\\\hline + & &
\\\hline & &
\\\hline
\end{array}
}\end{tinyrc}
\makebox[0pt][l]{\hbox{\footnotesize \ \ 231}}
\\
|\makebox[0pt][l]{$\reflection{2pt} s2$}
\\
\begin{tinyrc}{
\begin{array}{@{}|@{\,}c@{\,}|@{\,}c@{\,}|@{\,}c@{\,}|@{}}
\hline + &\phantom{+}&\phantom{+}
\\\hline & &
\\\hline & &
\\\hline
\end{array}
}\end{tinyrc}
\makebox[0pt][l]{\hbox{\footnotesize \ \ 213}}
\end{array}
\\
\reflection{-1pt} s2\diagdown\,
&
&
\,\diagup \reflection{-1pt} s1
\\
&
\begin{tinyrc}{
\begin{array}{@{}|@{\,}c@{\,}|@{\,}c@{\,}|@{\,}c@{\,}|@{}}
\hline \phantom{+}&\phantom{+}&\phantom{+}
\\\hline & &
\\\hline & &
\\\hline
\end{array}
}\end{tinyrc}
\\[7pt]
&
\hbox{\footnotesize 123}
\end{array}
$$
The right hand path from $321$ to $123$ is poptotic because there's only
one rc-graph at each stage. The left hand path is apoptotic because the
first rc-graph for $132$ has no offspring under $\mitosis_2$.%
\end{example}
Whether or not a path from $w_0$ to~$w$ is poptotic, breadth-first search
on the preimage tree (ordering the mitosis offspring as in
Proposition~\ref{prop:offspring}) yields a total order on $\rc(w)$. It
can be shown that {\em poptotic} total orders by breadth-first search are
linear extensions of the partial order on rc-graphs determined by chute
operations. Through heuristic arguments, computer calculations in small
symmetric groups, and the fact that $\LL_w$ is shellable
(Section~\ref{sub:cm}), we are convinced of the following.
\begin{conj} \label{conj:poptotic}
Poptotic total orders on $\rc(w)$ are shellings of $\LL_w$.
\end{conj}
To emphasize: shellability is not in question -- we will give shellings
in Section \ref{sub:cm}. The conjecture would just give some more
intuitive shellings than those we know.
It is conceivable that all of the apoptotic total orders are shellings,
too, but we hold out less hope for these.
\subsection{A combinatorial approach to mitosis}\label{sub:combin}%%%%%%%
It is possible to give a complete proof of Theorem~\ref{thm:mitosis}
based entirely on the BJS formula in Corollary~\ref{cor:BJS} and the
characterization of Schubert polynomials by divided differences, along
with elementary combinatorial properties of rc-graphs.
% No references to Gr\"obner bases, cohomology, $K$-theory, or even flag
% manifolds are required.
Before getting to a sketch of this proof, let us recast the definition of
intron from Section~\ref{sub:coarsen} directly in the language of
rc-graphs. We emphasize that no reference to Section~\ref{sub:coarsen}
is necessary here: these reformulations are independent. For
visualization purposes, recall that an elbow may be denoted by a
$\textelbow$, a dot, or an empty box in the diagrams.
\begin{defn} \label{defn:intron}
Let $D$ be a pipe dream and $i$ a fixed row index. Order the boxes in
the gene of $D$ (that is, rows $i$ and~$i+1$) as in the following
diagram:
$$
\begin{array}{l|c|c|c|c|c}
\\[-3ex]
\multicolumn{1}{l}{\mbox{}}
&\multicolumn{1}{c}{\petit{1}}
&\multicolumn{1}{c}{\petit{2}}
&\multicolumn{1}{c}{\petit{3}}
&\multicolumn{1}{c}{\petit{4}}
&\multicolumn{1}{c}{\cdots}
\\[.2ex]\cline{2-6}
\petit{i} & 1 & 3 & 5 & 7 &\toplinedots \\\cline{2-5}
\petit{i+1}& 2 & 4 & 6 & 8 & \\\cline{2-6}
\end{array}
$$
An \bem{intron} in this gene is a $2 \times k$ rectangle $C$ such that
\begin{thmlist}
\item
the first and last boxes in $C$ (the northwest and southeast corners) are
elbows;
\item
no elbow in $C$ is northeast or southwest of another elbow (but due north
or due south is okay); and
\item
the elbow with largest index before $C$ (if there is one) resides in
row~$i+1$, and the elbow with smallest index after $C$ (if there is one)
resides in row~$i$.
\end{thmlist}
\end{defn}
In the language of Section~\ref{sub:coarsen}, conditions~1 and~3
guarantee that $C$ is flanked by two exons (or possibly the start and/or
stop codon), while the middle condition says that $C$ contains no exons.
The intron mutation of Definition~\ref{defn:mutation} becomes here the
self-evident Lemma~\ref{lemma:intron}, below. Although it is irrelevant
for the present purpose, it may be worthwhile to note that the reason why
Lemma~\ref{lemma:intron} agrees with intron mutation (as per
Definition~\ref{defn:mutation}) restricted to maximal-degree squarefree
monomials $\not\in J_w$ comes from Theorem~\ref{thm:rc} and
Proposition~\ref{prop:pure}.
\begin{lemma} \label{lemma:intron}
Given an intron $C$, there is a unique intron $\tau_i C$ such that
\begin{thmlist}
\item
the sets of columns with exactly two crosses are the same in $C$ and
$\tau_i C$, and
\item
the number of crosses in row~$i$ of $C$ equals the number of crosses in
row~$i+1$ of $\tau_i C$, and conversely.
\end{thmlist}
The involution $\tau_i$ is called \bem{intron mutation}.\hfill$\Box$
\end{lemma}
For an example, simply replace every empty box by \cross and every
nonzero number by an elbow in each intron from Example~\ref{ex:gene}.
Another arbitrary example is:
\begin{equation} \label{eq:intron}
\begin{array}{c@{\ \qquad}c@{\qquad}c}
\begin{array}{lcccccccc}
\multicolumn{8}{c}{}\\[-6ex]
&\phperm&\phperm&\phperm&\phperm&\phperm&\phperm&\phperm&\phperm\\[3pt]
\petit{i} & \jr & \jr & \+ & \+ & \+ & \+ & \+ & \+
\\[-1pt]
\petit{i+1}& \+ & \jr & \+ & \jr & \+ & \jr & \jr & \jr
\end{array}
&
=
&
\begin{array}{l|c|c|c|c|c|c|c|c|}
\cline{2-9}
\petit{i} & & &\!+\!&\!+\!&\!+\!&\!+\!&\!+\!&\!+\!
\\\cline{2-9}
\petit{i+1}&\!+\!&\,\ &\!+\!& &\!+\!& & & \\\cline{2-9}
\end{array}
%
\\
\begin{array}{c}\\[-2ex]\quad\tau_i \downarrow\\[1ex]\end{array}
&&
\begin{array}{c}\\[-2ex]\quad\tau_i \downarrow\\[1ex]\end{array}
\\
%
\begin{array}{lcccccccc}
\multicolumn{8}{c}{}\\[-6ex]
&\phperm&\phperm&\phperm&\phperm&\phperm&\phperm&\phperm&\phperm\\[3pt]
\petit{i} & \jr & \jr & \+ & \jr & \+ & \jr & \jr & \+
\\[-1pt]
\petit{i+1}& \+ & \+ & \+ & \+ & \+ & \+ & \jr & \jr
\end{array}
&
=
&
\begin{array}{l|c|c|c|c|c|c|c|c|}
\cline{2-9}
\petit{i} & & &\!+\!& &\!+\!& &\,\ &\!+\!
\\\cline{2-9}
\petit{i+1}&\!+\!&\!+\!&\!+\!&\!+\!&\!+\!&\!+\!& & \\\cline{2-9}
\end{array}
\end{array}
\end{equation}
Here now is the rc-graph analogue of Lemma~\ref{lemma:involution}.
\begin{lemma} \label{lemma:sym}
For each $i$ there is an involution $\tau_i : \rc(w) \to \rc(w)$ such
that $\tau_i^2 = 1$, and for all $D \in \rc(w)$:
\begin{thmlist}
\item
$\tau_iD$ agrees with $D$ outside rows $i$ and $i+1$.
\item
$\start_i(\tau_iD) = \start_i(D)$, and $\tau_iD$ agrees with $D$ strictly
west of this column.
\item
$\ell^i_i(\tau_iD) = \ell^i_{i+1}(D)$,
\end{thmlist}
where $\ell^i_r(-)$ is the number of crosses in row $r$ that are east of
or in column $\start_i(-)$.
\end{lemma}
\begin{proof}
Let $D \in \rc(w)$. The union of all columns in the gene of $D$ that are
east of coincide with column $\start_i(D)$ can be written as a disjoint
union of $2 \times k$ rectangles, each of which is either an intron or
completely filled with crosses. Indeed, this follows from
(\ref{eq:pipestart}) and Definition~\ref{defn:intron}. Therefore the
lemma comes down to verifying that intron mutation preserves the
property of being in $\rc(w)$.
More accurately, mutation of a single intron in the gene of $D$
yields a pipe dream in $\rc(w)$. To verify this statement, one need only
check the routes of pipes intersecting the intron, and this is
straightforward: the numbers of crosses traversed horizontally by each
pipe are the same in $C$ and $\tau_i C$; similarly for the vertically
traversed crosses.%
\end{proof}
\begin{remark} \label{rk:sym}
The intron mutation illustrated in (\ref{eq:intron}) results from a
sequence of chute moves. This phenomenon is general, whenever the pipe
dream is an rc-graph: in any single intron of an rc-graph, either the
mutation or its inverse results from a sequence of chute moves. The
proof is omitted, but rests on the fact that the $2 \times 2$
configuration
%
$\begin{tinyrc}{
\begin{array}{@{}|@{\,}c@{\,}|@{\,}c@{\,}|@{}}
\hline \cdot & +
\\\hline + &\cdot
\\\hline
\end{array}
}\end{tinyrc}$
%
is disallowed in rc-graphs, so every intron in an rc-graph contains a
column with two elbows.
\end{remark}
\begin{trivlist}
\item{\it Sketch of combinatorial proof of Theorem~\ref{thm:mitosis}.\,}
The argument employs the description of mitosis in
Proposition~\ref{prop:offspring}, from which Lemma~\ref{lemma:rc} and
part~\ref{closed} of Lemma~\ref{lemma:BB} imply that $\mitosis_i(D)$
consists of rc-graphs for $ws_i$ whenever $D \in \rc(w)$. It follows
directly from the definitions that $\mitosis_i(D) \cap \mitosis_i(D') =
\nothing$ if $D \neq D'$ are rc-graphs for $w$. Thus it suffices to
prove that $\mitosis_i(\rc(w))$ has the same cardinality as $\rc(ws_i)$.
Using Lemma~\ref{lemma:sym}, one proves that
$$
\sum_{E \in \mitosis_i(D)} \xx^E \ + \sum_{E' \in \mitosis_i(\tau_i D)}
\xx^{E'} \quad = \quad \partial_i(\xx^D + \xx^{\tau_i D}).
$$
This is the same trick we applied in the context of lifted Demazure
operators $\ddem iw$ (Proposition~\ref{prop:lift}). Pairing off the
elements of $\rc(w)$ not fixed by $\tau_i$, as we did in the proof of
Proposition~\ref{prop:lift}, we conclude that
$$
\sum_{E \in \mitosis_i(\rc(w))} \xx^E \ \:=\:\ \partial_i \Bigl(\sum_{D
\in \rc(w)} \xx^D \Bigr) \ \:=\:\ \partial_i(\SS_w(\xx)) \ \:=\:\
\SS_{ws_i}(\xx) \ \:=\: \sum_{E \in \rc(ws_i)} \xx^E
$$
by Corollary~\ref{cor:BJS} and the definition of Schubert polynomial by
divided differences. Plugging in $(1, \ldots, 1)$ for $\xx$ implies that
$|\mitosis_i(\rc(w))| = |\rc(ws_i)|$, as desired.\hfill$\Box$
\end{trivlist}
\subsection{Shellability of subword complexes in Coxeter groups}%%%%%%%%%
\label{sub:cm}%%%%%%%%%%%
\comment{This subsection is complete except for
the percented-out comments after
Corollary~\ref{cor:cm}. As far as EM is
concerned, we should leave these other topics for
another time.}
Our goal in this subsection is Corollary~\ref{cor:shell}: the complexes
$\LL_w$ are shellable, and hence Cohen--Macaulay. Together with the
Gr\"obner basis theorem, this provides a new proof that Schubert
varieties are Cohen--Macaulay (Corollary~\ref{cor:cm}). We prove these
results by introducing a new class of vertex-decomposable simplicial
complexes, the subword complexes of Definition~\ref{defn:subword} and
Theorem~\ref{thm:shell}, and identifying the complexes $\LL_w$ as special
cases.
This subsection and the next deal with an arbitrary Coxeter system
$(\Pi,\Sigma)$ consisting of a group $\Pi$ and a set $\Sigma$ of
generators. See \cite{HumphCoxGrps} for background and definitions; the
applications to rc-graphs concern only the case where $\Pi = S_n$ and
$\Sigma$ consists of the simple reflections switching $i$ and $i+1$ for
$1 \leq i \leq n-1$.
\begin{defn} \label{defn:subword}
A \bem{word} of size~$m$ is an ordered list $Q = (\sigma_1, \ldots,
\sigma_m)$ of elements of~$\Sigma$. An ordered sublist $P$ of $Q$ is
called a \bem{subword} of $Q$.
\begin{thmlist}
\item
$Q$ \bem{represents} $\pi \in \Pi$ if the ordered product of the simple
reflections in $Q$ is a reduced decomposition for $\pi$.
\item
$Q$ \bem{contains} $\pi \in \Pi$ if some sublist of $Q$ represents $\pi$.
\end{thmlist}
The \bem{subword complex} $\Delta(Q,\pi)$ is the set of subwords $P
\subseteq Q$ whose complements $Q \minus P$ contain $\pi$.
\end{defn}
Sometimes we abuse notation and say that $Q$ is a word in $\Pi$.
% It might appear to be more ``correct'' to say "word in $\Sigma$", but I
% think it's better to suppress explicit mention of $\Sigma$ as much as
% possible. But that's just me. Search for the occurrences of ``word
% in'' if you want to change things. But see the Introduction, where it
% seems to make sense to say "word in a Coxeter group".
Note that $Q$ need not itself be a reduced expression. The following
lemma is immediate from the definitions and the fact that all reduced
expressions for $\pi \in \Pi$ have the same length.
\begin{lemma} \label{lemma:represents}
$\Delta(Q,\pi)$ is a pure simplicial complex whose facets are the
subwords $Q \minus P$ such that $P \subseteq Q$ represents
$\pi$.\hfill$\Box$
\end{lemma}
Subword complexes place rc-graphs in a context very close to that of
\cite{FSnilCoxeter} and \cite{FKyangBax}.
%, which motivated the definition of rc-graph \cite{BB} in the first place.
% how to write this so it doesn't look like [BB] introduced them?}
\begin{prop} \label{prop:rc}
$\LL_w$ is the join of a simplex with $\Delta(Q,w)$, where $\Pi = S_n$
and
\begin{eqnarray*}
Q &=& s_{n-1} \cdots s_2 s_1 s_{n-1} \cdots s_3 s_2 \cdots s_{n-1}
s_{n-2} s_{n-1},
\end{eqnarray*}
the \bem{lexicographically last} reduced expression for $w_0$, in which
$s_1 > s_2 > \cdots > s_{n-1}$ (this is the same order as in
Example~\ref{ex:lex}). In other words, an rc-graph for $w$ is a reduced
expression for $w$ that is a subword of the lex last long word.
\end{prop}
\begin{proof}
Use Lemma~\ref{lemma:subword}, Example~\ref{ex:rc}, and
Theorem~\ref{thm:rc}. The simplex has vertices corresponding to the
elbow joints on and below the main antidiagonal in~$[n]^2$.%
\end{proof}
\begin{defn} \label{defn:vertex}
Let $\Delta$ be a simplicial complex and $F \in \Delta$ a face.
\begin{thmlist}
\item
The \bem{deletion} of $F$ from $\Delta$ is $\del(F,\Delta) = \{G \in
\Delta \mid G \cap F = \nothing\}$.
\item
The \bem{link} of $F$ in $\Delta$ is $\link(F,\Delta) = \{G \in \Delta
\mid G \cap F = \nothing$ and $G \cup F \in \Delta\}$.
\end{thmlist}
$\Delta$ is \bem{vertex-decomposable} if $\Delta$ is pure and either
(1)~$\Delta = \{\nothing\}$, or (2)~for some vertex $v \in \Delta$, both
$\del(v,\Delta)$ and $\link(v,\Delta)$ are vertex-decomposable. A
\bem{shelling} of $\Delta$ is an ordered list $F_1, F_2, \ldots, F_t$ of
its facets such that $\bigcup_{j < i} F_j \cap F_i$ is a union of
codimension~1 faces of $F_i$ for each $i \leq t$. We say $\Delta$ is
\bem{shellable} if it is pure and has a shelling.
\end{defn}
\begin{remark} \label{rk:nothing}
The empty set $\nothing$ is a perfectly good face of $\Delta$,
representing the empty set of vertices. Thus the \bem{empty complex}
$\Delta = \{\nothing\}$, whose unique face has dimension $-1$, is to be
distinguished from the \bem{void complex} $\Delta = \{\}$ consisting of
no subsets of the vertex set. Writing out their reduced chain complexes,
the empty complex has integral reduced homology $\ZZ$ in dimension $-1$,
while the void complex has entirely zero homology. This distinction
reveals itself most prominently when taking the link of a facet (maximal
face) of $\Delta$: the result is the empty complex, not the void complex.
The link of $\nothing$ in $\Delta$ is just $\Delta$ itself, including
$\nothing \in \Delta$.%
\end{remark}
Provan and Billera \cite{BilleraProvan} introduced the notion of
vertex-decomposability and proved that it implies shellability (proof:
use induction on the number of vertices by first shelling
$\del(v,\Delta)$ and then shelling the cone from $v$ over
$\link(v,\Delta)$ to get a shelling of~$\Delta$). It is well-known that
shellability implies Cohen--Macaulayness \cite[Theorem~5.1.13]{BH}. Here
is our central observation concerning subword complexes.
\begin{thm} \label{thm:shell}
The subword complex $\Delta(Q,\pi)$ is vertex-decomposable. Therefore
subword complexes are Cohen--Macaulay and even shellable.
\end{thm}
\begin{proof}
Supposing that $Q = (\sigma, \sigma_2, \sigma_3, \ldots, \sigma_m)$, it
suffices to show that both the link and the deletion of $\sigma$ from
$\Delta(Q,\pi)$ are subword complexes. By definition, both consist of
subwords of $Q' = (\sigma_2, ..., \sigma_m)$. The link is naturally
identified with the subword complex $\Delta(Q',\pi)$. For the deletion,
there are two cases. If $\sigma \pi$ is longer than~$\pi$, then the
deletion of $\sigma$ equals its link because no reduced expression for
$\pi$ begins with~$\sigma$. On the other hand, when $\sigma \pi$ is
shorter than~$\pi$, the deletion is $\Delta(Q',\sigma\pi)$.%
\end{proof}
\begin{cor} \label{cor:shell}
$\LL_w$ is vertex-decomposable, and hence shellably Cohen--Macaulay.
\end{cor}
\begin{proof}
Theorem~\ref{thm:shell} and Proposition~\ref{prop:rc}.
\end{proof}
Among the known vertex decomposable simplicial complexes are the
% \bem{shifted complexes} \cite{DuvalShifting, HTstabAlgShift} and the
dual greedoid complexes \cite{BKL}, which include the matroid complexes.
Although subword complexes strongly resemble dual greedoid complexes, the
exchange axioms defining greedoids seem to be slightly stronger than the
exchange axioms for facets of subword complexes imposed by Coxeter
relations. In particular, the na\"\i ve ways to correspond these
complexes to dual greedoid complexes do not work, and we conjecture that
they are not in general isomorphic to dual greedoid complexes.
%\begin{remark} \label{rk:shell}
There seems to be little direct relation between our shellings of subword
complexes and the lexicographic shellings of intervals in the Bruhat
order by Bj\"orner and Wachs \cite{BWbruhCoxetShel}, even though the
results look superficially quite similar (see Section~\ref{sub:balls},
where we prove that subword complexes are balls or spheres). Their
results concern the Bruhat order, so their simplicial complexes are
independent of the reduced expressions involved, although their shellings
depend on such choices. In contrast, our results concern the weak Bruhat
order, where the reduced expressions involved form the substance of the
simplicial complexes, and fewer choices are left over for the shellings.
Nonetheless, comparison of the main results suggest that the Bruhat and
weak Bruhat orders ``feel'' somewhat similar.%
%\end{remark}
Fulton proved that $\ol X_w$ is Cohen--Macaulay in \cite{FulDegLoc}, but
he used the fact that Schubert varieties are Cohen--Macaulay
\cite{ramanathanCM} to do it. Here we can turn the tables, deriving a
new proof of the Cohen--Macaulayness in of Schubert varieties in flag
manifolds from an independent proof of Fulton's result.
\begin{cor} \label{cor:cm}
Every matrix Schubert variety $\ol X_w$, and hence every Schubert variety
$X_w \subseteq B\dom\gln$, is Cohen--Macaulay.
\end{cor}
\begin{proof}
The Cohen--Macaulayness of $\LL_w$ (Corollary~\ref{cor:shell}) implies
that of $\ol X_w$ by Theorem~\ref{thm:gb} and the flatness of Gr\"obner
deformation.
% See \cite{ConcaHerLadRatSing}, beginning of proof of 2.1 on p. 127
Now use Theorem~\ref{thm:equiv}.
\end{proof}
% \textbf{rational singularities? Frobenius split? Smooth in codim 2?
% What else should we show?}
%
% \begin{cor}
% $\ci w$ is $F$-injective.
% \end{cor}
% \begin{proof}
% Apply \cite[Corollary~2.2]{ConcaHerLadRatSing} to
% \comment{Corollary~\ref{cor:shell} and?}Theorem~\ref{thm:gb}.
% \end{proof}
\begin{remark} \label{rk:subword}
The vertex decomposition we use for matrix Schubert varieties has direct
analogues in the Gr\"obner degenerations and formulae for Schubert
polynomials. Consider the following sequence $>_1,>_2,\ldots,>_{n^2}$ of
partial term orders, where $>_i$ is lexicographic in the first $i$ matrix
entries snaking from northeast to southwest one row at a time, and treats
all remaining variables equally. The order $>_{n^2}$ is a total order;
this total order is antidiagonal, and hence degenerates $\ol X_w$ to the
subword complex by Theorem~\ref{thm:gb}. Each $>_i$ gives a degeneration
of $\ol X_w$ to a bunch of components. Each of these components
degenerates at $>_{n^2}$ to its own subword complex.
If we study how a component at stage $i$ degenerates into components at
stage $i+1$, by degenerating both using $>_{n^2}$, we recover the vertex
decomposition for the corresponding subword complex.
Note that these components are {\em not} always matrix Schubert
varieties; the set of rank conditions involved does not necessarily
involve only upper left submatrices. We do not know how general a class
of determinantal ideals can be tackled by partial degeneration of matrix
Schubert varieties, using antidiagonal partial term orders.
However, if we degenerate using the partial order $>_n$ (order just the
first row of variables), the components {\em are} matrix Schubert
varieties, except for the fact that the minors involved are all shifted
down one row. This gives an inductive formula for Schubert polynomials,
which already appears in Section~1.3 of \cite{BJS}.
\end{remark}
\subsection{Subword complexes are balls or spheres}\label{sub:balls}%%%%%
Knowing now that subword complexes in Coxeter groups are shellable, we
are able to prove a much more precise statement. Our proof technique
requires a certain deformation of the group algebra of a Coxeter group.
\begin{defn} \label{defn:prod}
Let $R$ be a commutative ring, and $\cD$ a free $R$-module with basis
$\{e_{\pi} \mid \pi \in \Pi\}$. Defining a multiplication on $\cD$ by
\begin{eqnarray} \label{eq:prod}
e_\pi e_\sigma &=& \left\{\begin{array}{@{\,}ll}
e_{\pi\sigma} & \hbox{if } \length(\pi\sigma) > \length(\pi)\\
e_{\pi} & \hbox{if } \length(\pi\sigma) < \length(\pi)
\end{array}\right.
\end{eqnarray}
for $\sigma \in \Sigma$ yields the \bem{Demazure algebra} of
$(\Pi,\Sigma)$ over $R$. Define the \bem{Demazure product} $\delta(Q)$
of the word $Q = (\sigma_1, \ldots, \sigma_m)$ by $e_{\sigma_1} \cdots
e_{\sigma_m} = e_{\delta(Q)}$.
\end{defn}
\begin{example} \label{ex:dem}
When $\Pi = S_n$ and $\Sigma$ is the set of simple reflections $s_1,
\ldots, s_{n-1}$, the algebra $\cD$ is generated over $R$ by the usual
Demazure operators $\dem i$ (hence the name ``Demazure algebra''). In
this case, the fact that $\cD$ is an associative algebra with given free
$R$-basis follows from the considerations in
Section~\ref{sub:schubintro}.
\end{example}
In general, the fact that the equations in~(\ref{eq:prod}) define an
associative algebra is the special case of
\cite[Theorem~7.1]{HumphCoxGrps} where all of the `$a$'~variables
equal~$1$ and all of the `$b$'~variables are zero. Observe that the
ordered product of a word equals the Demazure product if the word is
reduced. Here are some basic properties of Demazure products, in which
the `$\geq$' and `$>$' signs denote the Bruhat partial order on $\Pi$
\cite[Section~5.9]{HumphCoxGrps}.
\begin{lemma} \label{lemma:dem}
Let $P$ be a word in $\Pi$ and let $\pi \in \Pi$.
\begin{thmlist}
\item \label{geq}
The Demazure product $\delta(P)$ is $\geq \pi$ if and only if $P$
contains~$\pi$.
\item \label{=}
If $\delta(P) = \pi$, then every subword of $P$ containing $\pi$ has
Demazure product~$\pi$.
\item \label{>}
If $\delta(P) > \pi$, then $P$ contains a word $T$ representing an
element $\tau > \pi$ satisfying $|T| = \length(\tau) = \length(\pi)+1$.
\end{thmlist}
\end{lemma}
\begin{proof}
If $P' \subseteq P$ and $P'$ contains $\pi$, then $P'$ contains
$\delta(P')$ and $\pi = \delta(P) \geq \delta(P') \geq \pi$, proving
part~\ref{=} from part~\ref{geq}. Choosing any $\tau \in \Pi$ such that
$\length(\tau) = \length(\pi) + 1$ and $\pi < \tau \leq \delta(P)$ proves
part~\ref{>} from part~\ref{geq}.
Now we prove part~\ref{geq}. Suppose $\pi' = \delta(P) \geq \pi$, and
let $P' \subseteq P$ be the subword obtained by reading $P$ in order,
omitting any reflections along the way that do not increase length. Then
$P'$ represents $\pi'$ by definition, and contains $\pi$ because any
reduced expression for $\pi'$ contains a reduced expression for $\pi$.
On the other hand, suppose $T \subseteq P$ represents $\pi$, with $\tau
\in \Sigma$ being the last element of $T$. Use induction on $|P|$ as
follows. If $\tau$ is also the last element of $P$, then $\delta(P
\minus \tau) = \delta(P)\tau < \delta(P)$, so $\pi\tau \leq
\delta(P)\tau$ and thus $\pi \leq \delta(P)$. If $\tau$ isn't the last
element in $P$, then $\pi \leq \delta(P \minus \tau) = \delta(P)$
already.
\end{proof}
%\begin{remark} \label{rk:weyl}
%When $\Pi$ is a Weyl group, there is a geometric reason for
%parts~\ref{geq} and~\ref{=} of Lemma~\ref{lemma:dem}.
%Let $P_i$ denote the group of invertible matrices whose entries
%above the main diagonal are zero, except possibly the $(i,i+1)$-entry.
%To a word $Q$, one associates a {\em Bott-Samelson manifold}
%$B \ P_{\sigma_1} \times_B P_{\sigma_2} \times_B \ldots \times_B P_{\sigma_m}$,
%which has a natural map to $\fln$. It is easy to show the image is the
%Schubert variety $X_{\delta(Q)}$. Any
%
%
% \textbf{please
%insert Bott-Samelson reason here.}
%\end{remark}
\begin{lemma} \label{lemma:T}
Let $T$ be a word in $\Pi$ and $\pi \in \Pi$ such that $|T| =
\length(\pi) + 1$.
\begin{thmlist}
\item \label{atmost2}
There are at most two elements $\sigma \in T$ such that $T \minus \sigma$
represents~$\pi$.
\item \label{atleast2}
If $\delta(T) = \pi$, then there are two distinct $\sigma \in T$ such
that $T \minus \sigma$ represents~$\pi$.
\item \label{distinct}
If $T$ represents $\tau > \pi$, then $T \minus \sigma$ represents~$\pi$
for exactly one $\sigma \in T$.
\end{thmlist}
\end{lemma}
\begin{proof}
Part~\ref{atmost2} is obvious if $|T| \leq 2$, so choose elements
$\sigma_1,\sigma_2,\sigma_3 \in T$ in order of appearance, and let $T_1
\sigma_1 T_2\sigma_2 T_3\sigma_3 T_4 = T$ be the resulting partition
of~$T$ into connected segments. If two of the words $T \minus \sigma_i$
represent $\pi$, then assume $T \minus \sigma_3$ represents $\pi$ (by
reversing $T$ and replacing $\pi$ with $\pi^{-1}$ if necessary). Letting
$\tau_4$ be the ordered product of $T_4$, then $P := T \minus (\sigma_3
T_4) = T_1 \sigma_1 T_2 \sigma_2 T_3$ represents $\pi \tau_4^{-1}$. That
$P \minus \sigma_1$ and $P \minus \sigma_2$ can't both
represent~$\pi\tau_4^{-1}$ is a consequence of the exchange condition
\cite[Theorem~5.8]{HumphCoxGrps}, so multiplying on the right by
$\sigma_3 \tau_4$ yields the result.
In part~\ref{atleast2}, $\delta(T) = \pi$ means there is some $\sigma \in
T$ such that (i)~$T = T_1 \sigma T_2$; (ii)~$T_1 T_2$ represents~$\pi$,
and (iii)~$\tau_1 > \tau_1 \sigma$, where $T_1$ represents~$\tau_1$.
% Note that $\sigma \tau_2 > \tau_2$, since $T_1 T_2$ is a reduced
% expression.
Omitting some $\sigma'$ from $T_1 \sigma$ leaves a reduced expression for
$\tau_1 \sigma$. Thus $T \minus \sigma'$ represents $\pi$, and $\sigma'$
can't be the original $\sigma$, by~(iii). Part~\ref{distinct} is the
exchange condition.%
\end{proof}
\begin{remark} \label{rk:atmost2}
When $\Delta(Q,\pi)$ is $\LL_w$ (Proposition~\ref{prop:rc}), there is a
``pipe dream-theoretic'' reason for part~\ref{atmost2} of
Lemma~\ref{lemma:T}. Let $D$ be an rc-graph for $w$ with no $\cross$ at
$(i,j)$, and let $D_+$ be the pipe dream resulting by adding the extra
$\cross$ at $(i,j)$. This new $\cross$ in $D_+$ switches the
destinations of the pipes in $D$ squiggling through $(i,j)$. In order to
get an rc-graph {\em for the same $w$} out of $D_+$, we must reset the
destinations of these pipes by removing a $\cross$. Since the two pipes
in question crossed at most once in $D$, they can cross at most twice
in~$D_+$.%
\end{remark}
\begin{lemma} \label{lemma:manifold}
Suppose every codimension~1 face of a shellable simplicial
complex~$\Delta$ is contained in at most two facets. Then $\Delta$ is a
topological manifold-with-boundary that is homeomorphic to either a ball
or a sphere. The facets of the topological boundary of~$\Delta$ are the
codimension~1 faces of~$\Delta$ contained in exactly one facet
of~$\Delta$.
\end{lemma}
\begin{proof}
Let $B_d$ and $B_d'$ be homeomorphic to $d$-dimensional balls. If $h :
B_{d-1} \to B_{d-1}'$ is a homeomorphism between $(d-1)$-balls in their
boundaries, then $B_d \cup_h B_d'$ is again homeomorphic to a $d$-ball.
On the other hand, if $h : \partial B_d \to \partial B_d'$ is a
homeomorphism between their entire boundaries, then $B_d \cup_h B_d'$ is
homeomorphic to a $d$-sphere. Now use induction on the number of facets
of $\Delta$;
%, since any union of codimension~1 faces of a $d$-simplex is
%homeomorphic to either a $(d-1)$-ball or the boundary $(d-1)$-sphere.
if the $d$-sphere case is reached, the assumption on codimension~1 faces
of $\Delta$ implies that the shelling must be complete.%
\end{proof}
Of course, the shellable complexes that interest us are the subword
complexes.
\begin{thm} \label{thm:ball}
The subword complex $\Delta(Q,\pi)$ is a either a ball or a sphere. A
face $Q \minus P$ is in the boundary of $\Delta(Q,\pi)$ if and only if
$P$ has Demazure product $\delta(P) \neq \pi$.%
\end{thm}
\begin{proof}
That every codimension~1 face of $\Delta(Q,\pi)$ is contained in at most
two facets is the content of part~\ref{atmost2} in
Lemma~\ref{lemma:T}, while shellability is Theorem~\ref{thm:shell}.
This verifies the hypotheses of Lemma~\ref{lemma:manifold} for the first
sentence of the Theorem.
If $P$ has Demazure product $\neq \pi$, then $\delta(P) > \pi$ by
part~\ref{geq} of Lemma~\ref{lemma:dem}. Choosing $T$ as in part~\ref{>}
of Lemma~\ref{lemma:dem}, we find by part~\ref{distinct} of
Lemma~\ref{lemma:T} that $Q \minus T$ is a codimension~1 face contained
in exactly one facet of $\Delta(Q,\pi)$. Thus, using
Lemma~\ref{lemma:manifold}, we conclude that $Q \minus P \subseteq Q
\minus T$ is in the boundary of $\Delta(Q,\pi)$.
If $\delta(P) = \pi$, on the other hand, part~\ref{atleast2} of
Lemmas~\ref{lemma:dem} and~\ref{lemma:T} say that every codimension~1
face $Q \minus T \in \Delta(Q,\pi)$ containing $Q \minus P$ is contained
in two facets of $\Delta(Q,\pi)$. Lemma~\ref{lemma:manifold} says each
such $Q \minus T$ is in the interior of $\Delta(Q,\pi)$, whence $Q \minus
P$~must itself be an interior face.%
\end{proof}
\begin{cor} \label{cor:ball}
The initial complex $\LL_w$ is the join of a simplex with either a ball
or a sphere.
\end{cor}
\begin{proof}
Theorem~\ref{thm:ball} and Proposition~\ref{prop:rc}.
\end{proof}
\subsection{Combinatorics of Grothendieck polynomials}\label{sub:links}%%
For many applications, substituted Hilbert numerators $\KK(\1-\xx)$ are
more natural than the usual versions, being in some sense more geometric.
This is the essence behind the definition of multidegree from the Hilbert
numerator. When $\KK = \GG_w$ is a Grothendieck polynomial, for
instance, the Schubert polynomial~$\SS_w(\xx)$ is the sum of the lowest
degree terms in~$\GG_w(\1-\xx)$ by Lemma~\ref{lemma:schubert}, and we
have seen how these count rc-graphs. The goal of this section is the
analogue for \hbox{$\GG_w(\1-\xx)$} of Corollary~\ref{cor:BJS}
for~$\SS_w(\xx)$ due to Fomin and Kirillov \cite{FKgrothYangBax}.
However, as in Sections~\ref{sub:cm} and~\ref{sub:balls} (whose notation
we retain), the result for general subword complexes specializes to pipe
dreams. Our arguments are based on standard tools from combinatorial
commutative algebra, and we assume the notation of
Section~\ref{sub:alex}.
If $J \subseteq \kk[\zz]$ is a squarefree monomial ideal with zero set
$\LL$, then recall that the \bem{Alexander dual ideal} is defined as
$J^\star = \<\zz^{D_L} \mid L \subseteq \LL$ is a coordinate
subspace$\>$. Viewing the collection of coordinate subspaces in $\LL$ as
a simplicial complex, the minimal generators of $J^\star$ are therefore
$\{\zz^{D_L} \mid L \in \LL$ is a facet$\}$; smaller faces $L$ have
larger sets $D_L$, so they yield monomials $\zz^{D_L}$ that aren't
minimal generators of $J^\star$.
Each subword complex $\Delta(Q,\pi)$ determines a squarefree monomial
ideal $J \subseteq \kk[\zz]$, where the variables $z_1, \ldots, z_m$
correspond to the elements in $Q = (\sigma_1, \ldots, \sigma_m)$. The
components of the zero set of~$J$ correspond both to the facets of
$\Delta(Q,\pi)$ as well as to the generators of the Alexander dual
ideal~$J^\star$. Since $J^\star$ is $\ZZ^m$-graded, it has a minimal
$\ZZ^m$-graded free resolution
\begin{equation} \label{eq:freeres}
0 \from J^\star \from E_0 \from E_1 \from\cdots\from E_m \from 0,\qquad
E_i = \bigoplus_{P \subseteq Q} \kk[\zz](-\deg \zz^P)^{\beta_{i,P}},
\end{equation}
where $\beta_{i,P}$ is the \bem{$i^\th$ Betti number} of $J^\star$ in
$\ZZ^m$-graded degree $\deg\zz^P$.
Recall Hochster's formula \cite[p.~45]{MP} for the $\ZZ^m$-graded Betti
numbers of $J^\star$ in terms of reduced homology of $\Delta(Q,\pi)$:
\begin{eqnarray} \label{eq:hoc}
\beta_{i,P} &=& \dim_\kk\HH_{i-1}(\link(Q \minus P,\,\Delta(Q,\pi));\kk).
\end{eqnarray}
We employ this formula to calculate the Hilbert series of $J^\star$. For
notation, the monomial $\zz^P$ is short for $\prod_{\sigma_j \in P} z_j$
when $P$~is a subword of~$Q$. Note that the sum of the lowest degree
terms in Lemma~\ref{lemma:links} is $\sum \zz^P$ taken over $P$
representing~$\pi$.
\begin{lemma} \label{lemma:links}
If $J$ is the Stanley-Reisner ideal of $\Delta(Q,\pi)$ and $\ell =
\length(\pi)$, then
\begin{eqnarray*}
\KK(J^\star; \zz) &=& \sum_{\delta(P)=\pi} (-1)^{|P| - \ell} \zz^P
\end{eqnarray*}
is the numerator Hilbert series of the Alexander dual ideal,
where the denominator is $\prod_{j = 1}^m (1-z_j)$, as usual.
\end{lemma}
\begin{proof}
Let $Q \minus P \in \Delta(Q,\pi)$, so $P \subseteq Q$ contains $\pi$.
By Theorem~\ref{thm:ball}, either $\link(Q \minus P,\Delta(Q,\pi))$ is
contractible (if $\delta(P) \neq \pi$), or it is a sphere of dimension
\begin{eqnarray*}
\dim \Delta(Q,\pi) - |Q \minus P| &=& (|Q| - \ell - 1) - |Q\minus P|\
\:=\ \: |P| - \ell - 1
\end{eqnarray*}
(if $\delta(P) = \pi$), where a sphere of dimension~$-1$ is taken to mean
the empty complex~$\{\nothing\}$ having nonzero reduced homology in
dimension~$-1$ (Remark~\ref{rk:nothing}). Therefore $\HH_{i-1}\link(Q
\minus P,\Delta(Q,\pi))$ is zero unless $\delta(P) = \pi$ and $i = |P| -
\ell$. Now apply~(\ref{eq:hoc}) to~(\ref{eq:freeres}), and use
$H(J^\star; \zz) = \sum_j(-1)^j H(E_j; \zz)$.%
\end{proof}
Of course, our goal is to get information about Grothendieck polynomials,
which come from the Hilbert numerator of $J$ rather than $J^\star$.
However, these two are intimately related, as the next result
demonstrates. It holds more generally for the ``squarefree modules'' of
Yanagawa \cite{Yan}, as shown in \cite[Theorem~4.36]{Mil3}. A
$\ZZ$-graded version of Proposition~\ref{prop:inversion} was proved by
Terai for squarefree ideals using some calculations involving $f$-vectors
of simplicial complexes \cite[Lemma~2.3]{Ter}. The $\ZZ^m$-grading here
simplifies the proof.
\begin{prop}[Alexander inversion formula] \label{prop:inversion}
$\!$For any squarefree monomial ideal $J \subseteq \kk[\zz]$, we have
$\KK(\kk[\zz]/J;\zz) = \KK(J^\star;\1-\zz)$.
\end{prop}
\begin{proof}
See~(\ref{eq:J}) in Section~\ref{sub:alex} for the Hilbert series of
$\kk[\zz]/J$. On the other hand, the Hilbert series of $J^\star$ is the
sum of all monomials $\zz^\bb$ divisible by $\zz^{D_L}$ for some $L \in
\LL$, that is for $\bb \in \ZZ^m$ having support $D_L$ for some $L \in
\LL$:
\begin{equation} \label{eq:Jstar}
H(J^\star;\zz)\ \:=\ \:\sum_{L \in \LL} \prod_{i \in D_L}{z_i \over
1-z_i} \ \:=\ \:\sum_{L \in \LL} \frac{\prod_{i \in D_L}(z_i) \prod_{i
\not\in D_L}(1-z_i)}{\prod_{i=1}^m(1-z_i)}.
\end{equation}
Now compare the last expressions of (\ref{eq:J}) and~(\ref{eq:Jstar}).
\end{proof}
\begin{thm} \label{thm:links}
If $J$ is the Stanley-Reisner ideal of $\Delta(Q,\pi)$ and $\ell =
\length(\pi)$, then
\begin{eqnarray*}
\KK(\kk[\zz]/J; \zz) &=& \sum_{\delta(P)=\pi} (-1)^{|P| - \ell}
(\1-\zz)^P,
\end{eqnarray*}
where $(\1-\zz)^P = \prod_{i \in P}(1-z_i)$.
\end{thm}
\begin{proof}
Proposition~\ref{prop:inversion} and Lemma~\ref{lemma:links}.
\end{proof}
Special cases of Theorem~\ref{thm:links} and \cite[Theorem~2.3 and
p.~190]{FKgrothYangBax} agree. Their notation differs from ours:
substituting \mbox{$y_p \from 1-y_p^{-1}$} and $x_q \from 1-x_q$ in their
polynomial $\mathfrak L^{(-1)}_w(y,x)$ yields what we have called
$\GG_w(\xx,\yy)$. Therefore, in the following corollary, we write
$\GG_w(\1-\xx,\1-\yy^{-\1})$ for the polynomial obtained by substituting
$x_q \from 1-x_q$ and $y_p \from 1-y_p^{-1}$ into $\GG_w(\xx,\yy)$.
% instead of the geometrically motivated exponential weight $x_q/y_p$ for
% $z_{qp}$ (see Example~\ref{ex:grading}), they use $x_q y_p$.
% $\GG_{w_0}(\xx,\yy) = \prod_{i+j \leq n}(x_i+y_j-x_iy_j)$
% in \cite{FKyangBax}, but the operator $\pi_i^{(-1)}$ there is Alexander
% dual to the Demazure operator $\dem i$ here.
\begin{cor}[\cite{FKgrothYangBax}] \label{cor:links}
The double Grothendieck polynomial $\GG_w(\xx,\yy)$ satisfies
\begin{eqnarray*}
% \GG_w(\xx,\yy) &=& \sum_{\delta(D) = w}\prod_{(q,p)\in D} (-1)^{|D| -
% \ell} (1 - x_qy_p)\\
\GG_w(\1-\xx,\1-\yy^{-\1}) &=& \sum_{\delta(D) = w}\prod_{(q,p)\in D}
(-1)^{|D| - \ell} (x_q + y_p - x_q y_p),
\end{eqnarray*}
where\/ $\length(w) = \ell$. The version for single Grothendieck
polynomials reads
\begin{eqnarray*}
\GG_w(\1-\xx) &=&
% \DD_w(\zz)|_{z_{qp} = x_q}\ \:=\ \:
\sum_{\delta(D)=w} (-1)^{|D| - \ell} \xx^D.
\end{eqnarray*}
\end{cor}
\begin{proof}
Any pipe dream in $[n]^2$ having Demazure product $w \in S_n$ must be
contained in $D_0$ (from Example~\ref{ex:pipe}) by part~\ref{geq} of
Lemma~\ref{lemma:dem}. Therefore, Proposition~\ref{prop:rc} says $\LL_w
= \Delta(Q,w)$, where $Q$~is the word whose pipe dream fills $[n]^2$ with
crosses. Apply Theorem~\ref{thm:links} to this subword complex, so that
$J = J_w$ by definition. Specializing $z_{qp}$ to $x_q y_p$ yields the
double version after calculating $1-(1-x_q)(1-y_p) = x_q+y_p-x_q y_p$,
while the single version follows trivially.%
\end{proof}
Note that the sum of lowest degree terms in $\GG_w(\1-\xx,\1-\yy^{-\1})$
equals $\SS_w(\xx,-\yy)$, which has $(x_q + y_p)$ in place of the
difference $(x_q - y_p)$ appearing in Corollary~\ref{cor:doubSchub}. The
single version in Corollary~\ref{cor:BJS} is derived unchanged from
Corollary~\ref{cor:links}.
% mention Alexander dual definition of isobaric divided difference to
% justify substituting $\1-\xx,\1-\yy$;
\begin{remark} \label{rk:ER}
Corollary~\ref{cor:links} implies that the coefficients of
$\GG_w(\1-\xx)$ alternate. To be precise, there are polynomials
$\GG_w^{(d)}(\xx)$ with nonnegative coefficients such that
\begin{eqnarray} \label{eq:alternates}
\GG_w(\1-\xx) &=& \sum_{d \geq \ell} (-1)^{d-\ell}\GG_w^{(d)}(\xx),
\end{eqnarray}
where $\ell = \length(w)$. Since the results in \cite{FKgrothYangBax}
already imply this fact, it was one of our principal reasons for
conjecturing the Cohen--Macaulayness of $\LL_w$
(Corollary~\ref{cor:shell}) in the first place. The connection is
through the Eagon-Reiner theorem \cite{ER}:
\begin{quote}
A simplicial complex $\Delta$ is Cohen--Macaulay if and only if the
Alexander dual $J_\Delta^\star$ of its Stanley--Reisner ideal has
\bem{linear free resolution}, meaning that the differential in its
minimal $\ZZ$-graded free resolution over $\kk[\zz]$ can be expressed
using matrices filled with linear~forms.
\end{quote}
The numerator of the Hilbert series of any module with linear resolution
alternates as in~(\ref{eq:alternates}), so the Alexander inversion
formula (Proposition~\ref{prop:inversion}) and the Eagon-Reiner theorem
together say that (\ref{eq:alternates})~holds {\em if $\LL_w$ is
Cohen--Macaulay}. It would take suspiciously fortuitous cancellation to
have a squarefree monomial ideal whose Hilbert numerator behaves
like~(\ref{eq:alternates}) without the ideal actually having linear
resolution.
\end{remark}
% Here is a weird consequence of the characterization of
% $\KK(J_w^\star;\zz)$ via~$\DD_w(\zz)$.
Here is a weird consequence of the Demazure product characterization of
$\KK(J_w^\star;\zz)$.
\begin{por} \label{por:weird}
Each squarefree monomial $\zz^D$ in $\zz = (z_{qp})$ appears with nonzero
coefficient in the Hilbert numerator of exactly one ideal $J_w^\star$,
and its coefficient is $\pm 1$.
\end{por}
\begin{proof}
The permutation $w$ in question is $\delta(D)$, by
Lemma~\ref{lemma:links}.
\end{proof}
%\end{section}{Combinatorics of rc-graphs}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{section}{0}%\renewcommand\sectionname{Appendix}
\renewcommand{\thesection}{\Alph{section}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Appendix}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:appendix}
\subsection{Gradings, Hilbert series, and $K$-theory}\label{app:K-theory}
We review here some notions concerning $\ZZ^n$-gradings and Hilbert
series, and point out the connection to equivariant $K$-theory of vector
spaces with sufficiently nice torus actions. The algebraic side of this
material pervades the paper, while the more geometric aspects find
applications in Sections~\ref{sub:alex} and~\ref{sub:groth}. As in the
rest of this monograph, the field $\kk$ can have arbitrary
characteristic, and for convenience we assume that $\kk$ is algebraically
closed, although this hypothesis can be dispensed with by resorting to
sufficiently abstruse language.
Suppose a torus $T \cong (\kk^*)^n$ with weight lattice (character group)
$W \cong \ZZ^n$ acts linearly on the vector space $M \cong \kk^m$. Then
$M = \kk\cdot e_1 \oplus \cdots \oplus \kk\cdot e_m$ is a direct sum of
characters of $T$, say with weights $\aa_1, \ldots, \aa_m \in W$. If
$\zz = \{z_1, \ldots, z_m\} \in M^*$ is the basis dual to $e_1, \ldots,
e_m$, then the coordinate ring of $M$ is $\kk[\zz]$, which has a grading
induced by the action of~$T$:
$$
\zz^\bb = z_1^{b_1} \cdots z_m^{b_m} \quad \implies \quad \deg(\zz^\bb)
= \sum_{p=1}^m b_p\aa_p.
$$
Henceforth, we shall always assume that our torus actions have been given
in this form, so that $M$ is identified with $\kk^m$, and we choose an
identification $W = \ZZ^n$.
\begin{example} \label{ex:grading}
Our primary concern is the case $m = n^2$ and $M = \mn$, but in fact the
actions of four different tori play important roles. The largest,
$(\kk^*)^{n^2}$, scales all of the entries in a matrix independently; the
invertible $n \times n$ diagonal matrices $T$~act by multiplication on
the left; $T^{-1}$ is the same diagonal matrices, but acting by inverse
multiplication on the right; and $\kk^*$ acts by scaling all of the
entries in a matrix simultaneously.
The variables in $\kk[\zz]$ come in an array $\{z_{ij}\}_{i,j=1}^n$, and
their \bem{exponential weights} are
$$
\begin{array}{r|c@{\quad}c@{\quad}c@{\quad}c}
\hbox{torus action} & \kk^* & T & T \times T^{-1} & (\kk^*)^{n^2}
\\[.5ex]\hline
\hbox{exponential weight} & t\ & x_i & x_i/y_j\quad & z_{ij}\ \
\end{array}
$$
as Laurent monomials in the group algebra $\ZZ[$weight lattice$]$ over
the integers.
\end{example}
The set of weights $\aa \in \ZZ^n$ that are degrees of monomials in
$\kk[\zz]$ is an \bem{affine semigroup} (a finitely generated submonoid
of $\ZZ^n$), which we call the \bem{support semigroup}, with generators
$\aa_1, \ldots, \aa_m$. A $\kk[\zz]$-module $\Gamma$ is
\bem{$\ZZ^n$-graded} when $\Gamma = \bigoplus_{\aa \in \ZZ^n} \Gamma_\aa$
and $\zz^\bb \cdot \Gamma_\aa \subseteq \Gamma_{\aa+\deg(\bb)}$, where we
set $\deg(\bb) = \deg(\zz^\bb)$. We denote by $\Gamma(\aa)$ the
\bem{$\ZZ^n$-graded shift} of $\Gamma$ ``down by~$\aa$'', which is
defined by $\Gamma(\aa)_{\aa'} = \Gamma_{\aa+\aa'}$. All of the
homomorphisms $\Gamma \to \Gamma'$ of $\ZZ^n$-graded modules we use are
\bem{homogeneous}, meaning that the image of $\Gamma_\aa$ is contained in
$\Gamma'_\aa$ for all $\aa \in \ZZ^n$. For instance, multiplication by
$\zz^\bb$ induces an isomorphism between $\kk[\zz](-\deg(\bb))$ and the
principal ideal $\<\zz^\bb\>$.
It is a standard fact that the kernel and cokernel of a homomorphism of
$\ZZ^n$-graded modules is $\ZZ^n$-graded. In particular, every
$\ZZ^n$-graded module $\Gamma$ is a quotient of a $\ZZ^n$-graded free
module by a $\ZZ^n$-graded submodule. Furthermore, every $\ZZ^n$-graded
free module is isomorphic to a direct sum of rank one graded free modules
of the form $\kk[\zz](-\aa)$, whose generator is in degree $\aa \in
\ZZ^n$.
The ``sufficiently nice'' torus actions mentioned at the beginning of
this subsection are those satisfying the conditions in the next
definition. Such is the case with Example~\ref{ex:grading}, for
instance, where the support semigroup is $\NN^n$.
\begin{defn} \label{defn:grading}
The action of\/ $T$ on $\kk^m$ is \bem{positive} if
\begin{thmlist}
\item \label{p1}
the degrees $\aa_1, \ldots, \aa_m$ of $z_1, \ldots, z_m$ are nonzero; and
\item \label{p2}
the support semigroup has no nontrivial units (i.e.\ the real cone
spanned by $\aa_1, \ldots, \aa_m$ does not contain a linear subspace of
positive dimension).
\end{thmlist}
\end{defn}
The reason for defining positivity is that we want to work with
Hilbert series, as the following lemma allows.
\begin{lemma} \label{lemma:positivity}
The torus\/ $T$ acts positively on $\kk^m$ if and only if
$\dim_\kk(\Gamma_\aa)$ is finite for all finitely generated
$\kk[\zz]$-modules $\Gamma$ and all degrees $\aa \in \ZZ^n$.
\end{lemma}
\begin{proof}
If some variable $z_p$ has degree $\0 \in \ZZ^n$, then the subring
$\kk[z_p] \subseteq \kk[\zz]_\0$ has infinite dimension over $\kk$. If
the support semigroup has a unit $\aa = \deg(\zz^\bb)$, so that $-\aa =
\deg(\zz^{\bb'})$, then the subring $\kk[\zz^{\bb+\bb'}] \subseteq
\kk[\zz]_\0$ has infinite dimension over $\kk$. Therefore the finiteness
condition implies positivity.
On the other hand, suppose the action is positive. By condition~\ref{p2}
of positivity, there is a linear functional $\xi$ on $\ZZ^n$ which is
minimized on the support semigroup precisely at $\0$. By
condition~\ref{p1} of positivity, $\xi(\aa_p) > 0$ for $p = 1, \ldots,
m$. It follows that for each nonnegative $a \in \ZZ$, there are only
finitely monomials $\zz^\bb \in \kk[\zz]$ such that $\xi(\deg(\zz^\bb))
\leq a$. More generally, in any $\ZZ^n$-graded shift $\kk[\zz](-\bb)$
there are finitely many monomials with $\xi$-degree at most~$a$. Since
the finiteness condition is stable under finite direct sums and then
quotients, it holds for all finitely generated free modules and then all
finitely generated modules.
\end{proof}
Now we can define the \bem{Hilbert series}
\begin{eqnarray*}
H(\Gamma;\xx) &:=& \sum_{\aa \in \ZZ^n} \dim_\kk(\Gamma_\aa)\xx^\aa.
\end{eqnarray*}
of a finitely generated $\ZZ^n$-graded module $\Gamma$ over the
polynomial ring $\kk[\zz]$, in the presence of a positive $T$-action on
$\kk^m$. As it stands, this Hilbert series makes sense only as an
element in a direct product of copies of $\ZZ$, one for each $\aa \in
\ZZ^n$. To remedy this, let $A \subset \ZZ^n$ be the support semigroup
for the positive action of $T$ on $\kk^m$. The set of monomials $\xx^\aa
\neq 1$ in the semigroup ring $\ZZ[A] = \ZZ[\xx^{\aa_1}, \ldots,
\xx^{\aa_m}]$ generate a proper ideal. Completion with respect to this
ideal yields a ring $\ZZ[[A]]$ of power series supported on $A$. Let
$\ZZ[\xx^{\pm 1}]$ be the ring of Laurent polynomials in $\xx$, and
define the \bem{conical ring} for the positive action to be
\begin{eqnarray*}
\ZZ[\xx^{\pm 1}][[A]] &:=& \ZZ[\xx^{\pm 1}] \otimes_{\ZZ[A]} \ZZ[[A]].
\end{eqnarray*}
Its elements are those formal sums in which the Laurent monomials
appearing with nonzero (integer) coefficient are contained in a finite
union of translates of $A$ inside $\ZZ^n$. Note that $A$ need not
generate $\ZZ^n$ as a group---$A$ may even have dimension less
than~$n$---unless the action of $T$ is faithful.
\begin{example}
Consider what happens when $n=2$ and $z_1, \ldots, z_m$ all have weight
$(1,1)$. The grading on $\kk[\zz]$ is the usual $\ZZ$-grading, but
$(\alpha,\alpha^{-1}) \in T = \kk^* \times \kk^*$ acts trivially on
$\kk^m$, and $A = \NN \cdot (1,1) \subset \ZZ^2$ only spans a
subgroup of rank~$1$. The Hilbert series of finitely generated
$\ZZ^2$-graded modules can only have infinitely many terms along
finitely many rays pointing up diagonally with slope~$1$.
An example in a different direction is when $n = m = 2$, and the weights
of $z_1$ and $z_2$ are $\aa_1 = (2,0)$ and $\aa_2 = (0,2)$. In this
case, the kernel of the action of $T = (\kk^*)^2$ on $\kk^2$ is finite,
of order~$4$. There is a $\ZZ^2$-graded free module
$\kk[z_1,z_2]((-1,-1))$ generated in degree $(1,1)$, say, even though the
support of its Hilbert series is \hbox{disjoint from $A$}.%
\end{example}
For positive actions, the completeness of $\ZZ[[A]]$ implies that
$(1-\xx^{\aa_p})^{-1}$ is, for each $p$, represented by a well-defined
power series (i.e.\ convergent in the appropriate topology) in $\ZZ[[A]]
\subset \ZZ[\xx^{\pm 1}][[A]]$. This allows us to write
\begin{eqnarray*} \label{eq:hilb}
H(\kk[\zz](-\aa);\xx) &=& {\xx^{\aa}\over \prod_{p=1}^m
(1-\xx^{\aa_p})}\ \:\in \ZZ[\xx^{\pm 1}][[A]]
\end{eqnarray*}
for the Hilbert series of a rank one free module, and (given the Hilbert
syzygy theorem) proves the following proposition.
\begin{prop} \label{prop:numerator}
Let $T$ act positively on $\kk^m$. The Hilbert series of each finitely
generated $\ZZ^n$-graded $\kk[\zz]$-module $\Gamma$ is an element of the
conical ring $\ZZ[\xx^{\pm 1}][[A]]$. More precisely, there is a unique
\bem{Hilbert numerator} Laurent polynomial $\KK(\Gamma;\xx)$ with
\begin{eqnarray*}
H(\Gamma;\xx) &=& \frac{\KK(\Gamma;\xx)}{\prod_{p=1}^m
(1-\xx^{\aa_p})}.
\end{eqnarray*}
\end{prop}
Note that only certain very special elements of the conical ring are
actually Hilbert series of $\ZZ^n$-graded modules, and that Hilbert
series of ideals and quotients by them are always in $\ZZ[[A]]$ itself.
Our reason for caring about $\ZZ^n$-graded modules is that they are
global sections of $T$-equivariant sheaves on $\kk^m$, as we now show.
Recall that an \bem{equivariant sheaf} $\FF$ on $\kk^m$ is a sheaf along
with isomorphisms $\FF \to \alpha_* \FF$, for each $\alpha \in T$, that
are compatible with the group action. For instance, the isomorphism $\FF
\to (\alpha\beta)_* \FF$ is obtained by composing $\FF \to \beta_* \FF$
with $\beta_* \FF \to \alpha_*(\beta_*\FF)$.
\begin{lemma} \label{lemma:grading}
Let $\Gamma$ be a module over $\kk[\zz]$, and $\tilde\Gamma$ the
corresponding sheaf. Then $\ZZ^n$-gradings on $\Gamma$ correspond
bijectively to $T$-equivariant structures on $\tilde\Gamma$.
\end{lemma}
\begin{proof}
The direct sum decomposition of $\Gamma$ indexed by $\ZZ^n$ is the
splitting into isotypic components for the $T$-action. It is trivial to
verify that $\zz^\bb \Gamma_\aa \subseteq \Gamma_{\aa+\deg(\bb)}$. On
the other hand, the sheaf $\wt \Gamma$ associated to a $\ZZ^n$-graded module
$\Gamma$ comes with the torus action in which $\alpha \in T$ acts on the
global section $\sigma \in \Gamma_\aa$ via $\alpha \cdot \sigma =
\chi_\aa(\alpha) \sigma$, where $\chi_\aa : T \to \kk^*$ is the character
corresponding to $\aa$. Note that this action is compatible with
multiplication by $\zz^\bb$ because $\alpha \cdot \zz^\bb =
\chi_{\deg(\bb)} \zz^\bb$.
\end{proof}
Recall that the \bem{equivariant $K$-cohomology ring} $K^\circ_G(X)$ of a
variety $X$ with an action of a group $G$ is generated as a group by
isomorphism classes of $G$-equivariant vector bundles on $X$ modulo the
relations $[\EE]_G = [\EE']_G + [\EE'']_G$ determined by short exact
sequences $0 \to \EE' \to \EE \to \EE'' \to 0$, and has ring operations
induced by direct sum and tensor product. Allowing all equivariant
sheaves instead of just vector bundles yields the \bem{equivariant
$K$-homology group} $K_\circ^G(X)$, which is a module over $K^\circ_G(X)$
via direct sum and tensor product.
\begin{example} \label{ex:point}
When $X$ is a single point, an equivariant sheaf is the same thing as an
equivariant vector bundle: a representation of $G$. In this case
$K^\circ_G(\point)$ is the \bem{representation ring} of $G$. When $G =
T$ is a torus, $K^\circ_T(\point) \cong \ZZ[\xx^{\pm 1}]$ is generated as
an abelian group by the characters of $T$.
When $G = \{1\}$ is the trivial group, an equivariant sheaf is just a sheaf.
In this case, $K^\circ_{\{1\}}(X) = K^\circ(X)$ and $K_\circ^{\{1\}}(X) =
K_\circ(X)$ are called the \bem{ordinary $K$-groups}~of~$X$.%
\end{example}
If $X = \kk^m$ (or any smooth variety $M$), the Poincar\'e homomorphism
$K^\circ_G(X) \to K_\circ^G(X)$ sending an equivariant vector bundle to
its (automatically equivariant) sheaf of sections is an isomorphism.
More explicitly, an equivariant coherent sheaf $\FF$ has a finite
resolution by equivariant vector bundles,
\begin{equation} \label{eq:res}
0 \from \FF \from \EE_0 \from \EE_1 \from \cdots \from \EE_{m-1} \from
\EE_m \from 0,
\end{equation}
and we have $[\FF]_G = \sum_{p=0}^m (-1)^p [\EE_p]_G$ as a class in
$K^\circ_G(X)$.
\begin{prop} \label{prop:kclass}
$K^\circ_T(\kk^m) \cong \ZZ[\xx^{\pm 1}]$ for positive actions of $T$,
and the $K^\circ_T$-class of a $T$-equivariant coherent sheaf $\FF$ is
the Hilbert numerator $\KK(\Gamma(\FF);\xx)$.
\end{prop}
\begin{proof}
Every $\ZZ^n$-graded projective module is isomorphic to a direct sum of
free modules of the form $\kk[\zz](-\aa)$, by Nakayama's lemma (this
requires the positive grading if we are to avoid using the Quillen-Suslin
theorem \cite{Quillen,Suslin}). In other words, every equivariant vector
bundle on $\kk^m$ is a direct sum of equivariant line bundles of the form
$\OO_{\kk[\zz]} \otimes_\kk \kk_\aa$, where $\kk_\aa$ is the
$1$-dimensional $T$-representation with weight $\aa$. Since $\kk^m$ is
affine, the global section functor is exact, and it follows that every
short exact sequence of equivariant vector bundles is split. Thus
$K^\circ_T(\kk^m) \cong \ZZ[\xx^{\pm 1}]$.
Under this isomorphism, the $K^\circ_T$ class of $\OO_{\kk[\zz]}
\otimes_\kk \kk_\aa$ is just $\xx^\aa$, so the result holds for line
bundles by~(\ref{eq:hilb}). The result holds for equivariant vector
bundles on $\kk^m$ since these are all direct sums of equivariant line
bundles. Finally, the result holds for arbitrary equivariant coherent
sheaves by Lemma~\ref{lemma:grading} and the line after~(\ref{eq:res}).
\end{proof}
\begin{example} \label{ex:functorial}
Everything in this subsection is functorial with respect to the choice of
torus~$T$. In the situation of Example~\ref{ex:grading}, for example,
the inclusions of tori induce maps
$$
% | | |
\begin{array}{ccccccc}
\kk^* &\into& T &\into& T \times T^{-1} & \to & (\kk^*)^{n^2} \\
\ZZ[t^{\pm 1}]
&\otno&\ZZ[\xx^{\pm 1}]
&\otno&\ZZ[\xx^{\pm 1},\yy^{\pm 1}]
&\from&\ZZ[\zz^{\pm 1}]\\
t &\from& x_i &\from& x_i/y_j &\from& z_{ij}
\end{array}
$$
on equivariant $K$-theory of~$\mn$. These maps amount to \bem{coarsening
the grading}\/: every $\ZZ^{n^2}$-graded module is also
$\ZZ^{2n}$-graded; every $\ZZ^{2n}$-graded module is also $\ZZ^n$-graded,
and so on, via the maps between weight lattices.%
\end{example}
More generally, an arbitrary action of $T$ on $\kk^m$ is induced by a map
from $T$ to the torus $(\kk^*)^m$ scaling each of the variables $z_1,
\ldots, z_m$ separately. Taking the standard basis $\ee_1, \ldots,
\ee_m$ for $\ZZ^m$, positivity can be rephrased in terms of the resulting
map $\ZZ^m \to W$ of weight lattices by saying that the kernel intersects
$\NN^m$ trivially.
This perspective on $\kk[\zz]$ with its grading by $W$ leads naturally to
toric geometry in two forms. The extra information of a certain radical
monomial ideal makes $\kk[\zz]$ into a ``homogeneous coordinate ring''
\cite{Cox}. On the other hand, the extra information of the surjection
$\kk[\zz]$ onto the semigroup ring $\kk[{\mathcal A}]$, where ${\mathcal
A}$ is the image in $W$ of the standard basis in~$\ZZ^m$, has kernel
$I_{\mathcal A}$, which is a ``lattice ideal'' \cite{PSgeneric}.
\subsection{Term orders and Gr\"obner bases}\label{app:gb}%%%%%%%%%%%%%%%
% For the reader's convenience,
This appendix collects some standard definitions and facts surrounding
the theory of Gr\"obner bases. At the end, we point out where in this
monograph the various aspects of ``Gr\"obnerness'' turn up.
Resume the general notation $\kk[\zz] = \kk[z_1,\ldots,z_m]$ of
Appendix~\ref{app:K-theory}.
%and~\ref{app:degrees}.
A total order~`$>$' on the monomials in $\kk[\zz]$ is a \bem{term order}
if
\begin{eqnarray}
\label{eq:mult}
\zz^{\aa+\bb} > \zz^{\aa+\cc}
&\hbox{ whenever }
&\zz^\bb > \zz^\cc \quad (\aa,\bb,\cc \in \NN^m), \hbox{ and }\\
\label{eq:>1}
\zz^\bb\ > 1 \quad\
&\hbox{ whenever }
&\0 \neq \bb \in \NN^m.
\end{eqnarray}
The first of these two conditions is read ``the total order `$>$' is
\bem{multiplicative}.'' Together, (\ref{eq:mult}) and~(\ref{eq:>1})
imply that
\begin{eqnarray} \label{eq:artin}
\zz^{\aa+\bb} > \zz^\aa
&\hbox{ for all }
&\aa,\bb \in \NN^m \hbox{ with } \bb \neq \0,
\end{eqnarray}
and the following key property is a consequence.
\begin{lemma} \label{lemma:artinian}
Term orders are \bem{artinian}, meaning that every set of monomials
in~$\kk[\zz]$ possesses a (necessarily unique) minimal element. In
particular, every descending sequence of monomials stabilizes.
\end{lemma}
\begin{proof}
If $A$ is a set of monomials, then a finite subset $A' \subseteq A$
generates the ideal~$\$, because $\kk[\zz]$ is noetherian. The
smallest element in~$A'$ is minimal in~$A$ by~(\ref{eq:artin}).
\end{proof}
One of the best known term orders is the \bem{lexicographic} term order,
in which $\zz^\aa >_{\rm lex} \zz^\bb$ if the {\em earliest}\/ nonzero
coordinate of $\aa - \bb$ is {\em positive}. As B.~Sturmfels likes to
say, $z_1$ is {\em so}\/ expensive, that any monomial containing more
$z_1$ than another must automatically be larger in the total order. And
if two monomials have the same amount of $z_1$, then $z_2$ is the
expensive variable, and so on. Technically, there is a lexicographic
term order for each total ordering of the variables themselves; the lex
term order in this paragraph has $z_1 > z_2 > \cdots > z_m$.
The other standard term order with $z_1 > z_2 > \cdots > z_m$ is the
\bem{reverse lexicographic} term order, in which $\zz^\aa >_{\rm revlex}
\zz^\bb$ whenever $\sum a_i = \sum b_i$ and the {\em latest}\/ nonzero
coordinate of $\aa-\bb$ is {\em negative}. Again, as Sturmfels explains
it, $z_m$ is {\em so}\/ cheap, that any monomial containing more $z_m$
than another monomial of the same degree must automatically be smaller in
the total order.
Given a term order `$>$' (or indeed, any total order on monomials in
$\kk[\zz]$), every polynomial $f \in \kk[\zz]$ has an \bem{initial term}
$\IN_>(f)$, which we abbreviate to $\IN(f)$ when the term order `$>$' is
clear from context. The term $\IN(f)$ is defined to be $\lambda
\zz^\bb$, where $\zz^\bb$ is the largest monomial in the total order that
has nonzero coefficient in~$f$, and the nonzero coefficient on $\zz^\bb$
in $f$ is~$\lambda \in \kk^*$.
\begin{example} \label{ex:antidiag}
Consider two term orders on $\kk[\zz]$, where $\zz = (z_{ij})_{i,j=1}^n$:
\begin{enumerate}
\item
the reverse lexicographic term order that snakes its way from the
northwest corner to the southeast corner, $z_{11} > z_{12} > \cdots >
z_{1n} > z_{21} > \cdots > z_{nn}$; and
\item
the lexicographic term order that snakes its way from the northeast to
the southwest,
% $z_{1\,n} > z_{1\,n-1} > \cdots > z_{1\,1} > z_{2\,n} > \cdots >
% z_{n\,1}$.
$z_{1,n} > z_{2,n} > \cdots > z_{n,n} > z_{1,n-1} > \cdots >
z_{n-1,1} > z_{n,1}$.
\end{enumerate}
Let us verify the statement made after Definition~\ref{defn:Iw} to the
effect that for either term order, the initial term of any minor in the
generic matrix $Z = (z_{ij})$ is the antidiagonal term. For simplicity,
let's just check this for $\det(Z)$ itself.
Start with the revlex order. Every monomial in $\det(Z)$ contains a
variable from the last row. If this variable isn't~$z_{n1}$, then the
monomial is {\em so}\/ cheap that it can't be the initial term.
Therefore $z_{n1}$ divides $\IN_{\rm revlex}(\det(Z))$. Repeat the
argument on the submatrix of $Z$ obtained by crossing out its left column
and bottom row, and so on.
Now do the lex order. Every monomial in $\det(Z)$ contains a variable
from the last column. If this variable is~$z_{1n}$, then the monomial is
{\em so}\/ expensive that it must be larger than any monomial without
$z_{1n}$. Therefore $z_{1n}$ divides $\IN_{\rm lex}(\det(Z))$. Repeat
the argument on the submatrix of $Z$ obtained by crossing out its top row
and rightmost column, and so on. Observe that snaking the other way from
northeast to southwest (that is, across the top row first instead of down
the rightmost column first) produces the same result, by the symmetry
reflecting across the antidiagonal.
\end{example}
Algebraically, the distinction between term orders and arbitrary total
orders on monomials becomes clear when considering the \bem{initial
ideal} of an ideal $I \subseteq \kk[\zz]$, defined as \mbox{$\IN_>(I) :=
\{\IN_>(f) \mid f \in I\}$}. Again, we write $\IN(I)$ when the context
is clear. That $\IN(I)$ is indeed an ideal follows easily
from~(\ref{eq:mult}), (\ref{eq:>1}), and~(\ref{eq:artin}).
\begin{prop}
The monomials outside $\IN(I)$ form a $\kk$-vector space basis for
$\kk[\zz]/I$.
\end{prop}
\begin{proof}
The initial term of any nonzero $\kk$-linear combination of monomials
outside $\IN(I)$ lies outside~$\IN(I)$. Therefore, no such linear
combination can lie in~$I$, so the monomials outside $\IN(I)$ remain
linearly independent modulo~$I$.
Given a polynomial $f \in \kk[\zz]$, let $\max(f)$ be the largest
monomial of $\IN(I)$ with nonzero coefficient in~$f$. If no such
monomial exists, then $f$~lies in the span $S$ of monomials outside
$\IN(I)$. Otherwise, pick $g \in I$ with $\IN(g) = \max(f)$, and
subtract the appropriate scalar multiple of $g$ from $f$. This produces
a polynomial $f'$ that either lies in $S$ or has $\max(f) > \max(f')$.
This process must terminate by Lemma~\ref{lemma:artinian}.%
% Let $V$ be the $\kk$-span of the monomials outside~$\IN(I)$, and define
% $F$ to be the set of polynomials not having a representative in $V$
% modulo~$I$. For $f \in F$, let $\max(f)$ be the largest monomial of
% $\IN(I)$ with nonzero coefficient in~$f$. If $F$ is nonempty, then
% pick $f \in F$ with $\max(f)$ minimal, and choose $g \in I$ with
% $\IN(g) = \max(f)$. Subtracting the appropriate scalar times $g$
% from~$f$ either yields a polynomial $f' \in F$ with $\max(f) >
% \max(f')$, or an element of~$V$. The first possibility contradicts the
% minimality of~$\max(f)$, while the second contradicts $f \in F$.
\end{proof}
A set $\{f_1,\ldots,f_r\}$ of elements in~$I$ is called a \bem{Gr\"obner
basis} for~$I$ if $\IN(I) = \<\IN(f_1),\ldots,\IN(f_r)\>$ is generated by
the initial terms of the polynomials $f_1,\ldots,f_r$. Such finite sets
always exist because $\kk[\zz]$ is noetherian. Our main examples of
Gr\"obner bases are, of course, the minors in Theorem~\ref{thm:gb}. The
property of Gr\"obner bases that we exploit most is the following.
\begin{cor} \label{cor:flat}
Assume $\kk[\zz]$ is graded via a positive torus action. Then
$\kk[\zz]/I$ and $\kk[\zz]/\IN(I)$ have the same $\ZZ^n$-graded Hilbert
series.
\end{cor}
\begin{proof}
The two Hilbert series in question count the dimensions of vector spaces
having the same set of monomials for bases, namely the monomials outside
$\IN(I)$.%
\end{proof}
Various aspects of Gr\"obner bases and the passage to initial ideals
arise in this monograph. Roughly, they can be classified as algebraic,
combinatorial, and geometric, the same way as in the Introduction.
In the next appendix, Gr\"obner degeneration plays an algebraic role: the
preservation of Hilbert series (and therefore Hilbert numerators) under
taking initial ideals helps us define the notion of multidegree. This
occurs particularly in Proposition~\ref{prop:multideg}. Multidegrees,
which are significantly coarser invariants than Hilbert series, are one
of the two avenues by which Corollary~\ref{cor:flat} enters into the
proof of Theorem~\ref{thm:gb}, via Lemma~\ref{lemma:IN} in this case.
For the second (and more direct) application, we calculate the Hilbert
series of matrix Schubert varieties by finding the Hilbert series of
their initial ``antidiagonal'' ideals.
The calculation of the initial ideals' Hilbert series relies on actually
knowing the initial terms that generate them. In other words, we take
advantage of a specific Gr\"obner {\em basis}\/ rather than some abstract
Gr\"obner {\em degeneration}. The Gr\"obner basis consists of
geometrically meaningful polynomials (minors), whose initial terms
(antidiagonals) have crucial combinatorial properties. The analysis of
these antidiagonal terms occupies most of Section~\ref{sec:gb}; in
addition to the Hilbert series of Theorem~\ref{thm:gb}, it results also
in the purely combinatorial theorems forming the first half of
Section~\ref{sec:rc}.
Geometrically, Corollary~\ref{cor:flat} says that Gr\"obner degeneration
is flat, or in other words that the Gr\"obner degenerated monomial
subscheme defines the same class in equivariant $K$-theory. This
geometric perspective on Gr\"obner degeneration as an algebraic homotopy
is the motor behind our intuition and proof of the positivity of
coefficients in Schubert polynomials, thought of as representing the
cohomology classes of Schubert varieties in flag manifolds (see the
Introduction and the beginning Appendix~\ref{app:degrees}). It also
provides the backdrop for Section~\ref{sub:groth}, which identifies the
Grothendieck polynomials as $K$-classes of Schubert varieties in flag
manifolds.
As a final remark, it is possible to make sense of intial ideals with
respect to ``partially defined'' term orders, which fail to distinguish
between some monomials, but which admit refinements to honest term
orders. Such is the case in Remark~\ref{rk:subword}, for instance.
\subsection{Multidegrees of graded modules}\label{app:degrees}%%%%%%%%%%%
\def\hot{{\rm h.o.t.}}
\def\mult{{\rm mult}}
Resume the notation from Appendix~\ref{app:K-theory}, so $\kk[\zz]$ is a
$\ZZ^n$-graded polynomial ring in $m$ variables. Let $\Gamma$ be a
$\ZZ^n$-graded module. We shall define in this Appendix the multigraded
degree of $\Gamma$, in a way that generalizes the usual notion of degree
for $\ZZ$-graded ideals and subschemes of projective space. We assume in
this appendix that the action of the torus $T = (\kk^*)^n$ on $\kk^m$ is
positive, but see Remark~\ref{rk:positive}.
The source of our intuition is that the multidegree of the quotient
$\kk[\zz]/I$ for a homogeneous radical ideal~$I$ ought to be nothing more
than the $T$-equivariant cohomology class of the subvariety of~$I$, at
least when $\kk = \CC$. This can be made completely precise, although we
would need better machinery for producing equivariant cohomology classes
from subvarieties than the Vassiliev--Kazarian method in the next
appendix. The more powerful machinery would be forced by the fact that
$T$ usually fails to have finitely many orbits on $\CC^m$, so that
$T$-stable subvarieties will rarely be $(\CC^*)^m$-stable as would be
required by Corollary~\ref{cor:subvariety}.
In algebraic terms, this means that $I$ need not be a monomial ideal, so
we can't just specialize the natural $\ZZ^m$-graded degree of $I$ (we
shall determine what that is for nonreduced monomial ideals) to the
$\ZZ^n$-graded degree of~$I$. On the other hand, Gr\"obner deformation
always respects the $\ZZ^n$-grading: homogenizing a $\ZZ^n$-graded
polynomial with respect to some weight vector and then setting the
homogenizing variable to some constant always results in another
$\ZZ^n$-graded polynomial. Therefore, we can simply {\em define} the
multidegree of $I$ in terms of its initial ideal, having faith that
because Gr\"obner deformation preserves equivariant cohomology classes,
our defintion of multidegree remains independent of the term order (this
independence is proved in Proposition~\ref{prop:multideg}, below).
To begin with, recall the standard notion of \bem{multiplicity}
$\mult_X(\Gamma)$ of a module $\Gamma$ along an irreducible
subvariety~$X$ in $\kk^m$: it is the length of the largest submodule of
finite length in the localization of $\Gamma$ at the prime ideal of~$X$.
When $X$ has Krull dimension $\dim \Gamma$, this localization already has
finite length $\mult_X(\Gamma)$.
Our initial use of multiplicity will be for modules $\Gamma$ that are
direct sums of quotients by monomial ideals, in which case we can safely
assume that $X$ is a coordinate subspace \hbox{$L \subseteq \kk^m$}. We
mention this because the reader may know other more combinatorial
characterizations of multiplicity at monomial primes, for instance in
terms of `standard pairs'~\cite{STV}. Let $D_L$ be the subset of $\{1,
\ldots, m\}$ such that \hbox{$\$} is the ideal of
functions vanishing on~$L$.
Our main new ingredient for multidegrees looks a little strange at first
sight. Set
\begin{eqnarray} \label{eq:log}
\log_\xx(\zz^\bb) &=& \prod_{b_i \neq 0} \bigg(b_i \sum_{j=1}^n a_{ij}
x_j\bigg)
\quad\hbox{for any monomial}\quad \zz^\bb \in \kk[\zz].
\end{eqnarray}
The numbers $a_{ij}$ are, for each $i = 1,\ldots,m$, the components of
the degree \hbox{$\aa_i \in \ZZ^n$} of~$z_i$. The right hand side
of~(\ref{eq:log}) should be interpreted as an element in the ring
$H^*_T(\kk^m) = \sym^\spot_\ZZ(\ZZ^n)$, and could more ``correctly'' be
written as the element $\prod_{b_i \neq 0} b_i \deg(z_i) \in
\sym^c_\ZZ(\ZZ^n)$, where $c$ is the number of nonzero coordinates
of~$\bb$.
\begin{defn} \label{defn:geomdeg}
Let $J \subseteq \kk[\zz]$ be a monomial ideal with zero scheme $\LL
\subseteq \kk^m$. The ($\ZZ^n$-graded) \bem{geometric multidegree} of
$\LL$ (or of $\kk[\zz]/J$) is the sum
\begin{eqnarray*}
[\LL]_{\ZZ^n} &=& \sum\mult_L(\kk[\zz]/J) \cdot \log_\xx(\zz^{D_L})
\end{eqnarray*}
over all {\em reduced} subspaces $L \subseteq \LL$ of maximal dimension.
Suppose that $J_1,\ldots,J_r$ are monomial ideals with zero schemes
$\LL_1,\ldots,\LL_r$, and that $\aa_1,\ldots,\aa_r \in \ZZ^n$. The
\bem{geometric multidegree} of the module $\Gamma =
\bigoplus_{\ell=1}^r(\kk[\zz]/J_\ell) (\aa_\ell)$ is the sum $\sum
[\LL_\ell]_{\ZZ^n}$ over all $\ell$ such that $\dim(\LL_\ell) =
\dim(\Gamma)$.
\end{defn}
Note that geometric multidegrees ignore $\ZZ^n$-graded shifts. Before
going ahead and defining multidegrees of arbitrary $\ZZ^n$-graded
quotients by taking initial ideals, we have to know that the multidegree
is independent of the term order. This will be evident once we can read
the multidegree of a direct sum of monomial quotients of $\kk[\zz]$ off
of its Hilbert series (Proposition~\ref{prop:hilbdeg}). In order to
prove the desired result, we need to make a short foray
% digression
into combinatorial commutative algebra.
A $\ZZ^m$-graded $\kk[\zz]$-module $\Gamma$ is called
\bem{$\NN^m$-graded} if the graded pieces of $\Gamma$ are nonzero only in
degrees from $\NN^m$. Such is the case, for instance, for all monomial
ideals and quotients by them. An \bem{irreducible} monomial ideal is an
ideal generated by powers of the variables~$z_i$. For convenience, we
denote the ideal $\$ by $\mm^\bb$ whenever
$\bb \in \NN^m$. An \bem{irreducible resolution} of $\Gamma$ is an exact
sequence
$$
0 \to \Gamma \to W^0 \to W^1 \to \cdots
$$
in which $W^k = \bigoplus_\ell \kk[\zz]/\mm^{\bb_{k\ell}}$ is a direct
sum of quotients by irreducible monomial ideals.
\begin{lemma} \label{lemma:irredres}
Every $\NN^m$-graded $\kk[\zz]$-module $\Gamma$ has a finite irreducible
resolution in which $\dim W^k \leq \dim(\Gamma)$ for all $k \geq 0$.
\end{lemma}
\begin{proof}
There are a number of ways to prove the existence of finite irreducible
resolutions. One could apply the Alexander duality functor with respect
to some degree $\aa \in \ZZ^m$ to a free resolution of the Alexander dual
module $\Gamma^\aa$ \cite{Mil2}. Alternatively, one can take the
$\NN^m$-graded part of a $\ZZ^m$-graded injective resolution of $\Gamma$;
this is the way the proof goes in \cite{MilCM}, where irreducible
resolutions were first defined (for monomial ideals in semigroup rings).
The dimension inequality actually follows from the injective resolution
argument.
\end{proof}
Given a Laurent monomial $\xx^\aa$ for some $\aa \in \ZZ^n$, the rational
function $\prod_{j=1}^n(1-x_j)^{a_j}$ can be expanded as a well-defined
(i.e.\ convergent in the $\xx$-adic topology) formal power series
$\prod_{j=1}^n(1 - a_jx_j + \cdots)$ in~$\xx$. Doing the same for
\hbox{an arbitrary Laurent polynomial} $\KK(\xx)$ results in a power
series $\KK(\1-\xx)$ called a \bem{substituted Laurent polynomial}.
\begin{prop} \label{prop:hilbdeg}
If $J$ is a monomial ideal with zero scheme $\LL$, then the sum of all
lowest degree terms in $\KK(\kk[\zz]/J;\1-\xx)$ equals the geometric
multidegree~$[\LL]_{\ZZ^n}$. This sum is a homogeneous form (in the
usual sense) of degree $m - \dim(\LL)$ in~$\xx$.
\end{prop}
\begin{proof}
First we prove the result when $J = \mm^\bb$. Since the lowest free
resolution of this $J$ is a Koszul complex, the $\ZZ^m$-graded Hilbert
numerator of $\kk[\zz]/J$ is \hbox{$\prod_{b_i \neq 0} (1 - z_i^{b_i})$}.
The Hilbert numerator for the $\ZZ^n$-grading is therefore equal to
\hbox{$\prod_{b_i \neq 0} (1 - \xx^{b_i\aa_i})$}. Substituting $1-x_j$
for each occurrence of $x_j$ ($j = 1,\ldots,n$) yields
\begin{eqnarray}
\prod_{b_i \neq 0} \bigg(1 - \prod_{j=1}^n(1-x_j)^{b_i a_{ij}}\bigg)
&=& \nonumber
\prod_{b_i \neq 0} \bigg(1 - \prod_{j=1}^n(1-b_i a_{ij} x_j+\hot)\bigg)
\\&=& \nonumber
\prod_{b_i \neq 0} \bigg(\sum_{j=1}^n(b_i a_{ij} x_j)+\hot\bigg)
\\&=& \nonumber
\log_\xx(\zz^\bb) + \hot
\\&=& \label{eq:hot}
\bigg(\prod_{b_i \neq 0} b_i\bigg) \cdot \bigg(\prod_{b_i \neq 0}
\log_\xx(z_i)\bigg) + \hot,
\end{eqnarray}
as desired ($\hot$ denotes higher order terms in the $\xx$ variables).
We have used the fact that the multiplicity of $\kk[\zz]/J$ at its unique
associated prime is $\prod_{b_i \neq 0} b_i$.
Now let $J$ be an arbitrary monomial ideal, and consider an irreducible
resolution $W^\spot$ of $\kk[\zz]/J$ satisfying the conditions of
Lemma~\ref{lemma:irredres}. Substituting $\1-\xx$ for $\xx$ in the
Hilbert numerator of~$\kk[\zz]/J$ yields the finite alternating sum
\begin{eqnarray} \label{eq:NW}
\KK(\kk[\zz]/J;\1-\xx) &=& \sum_{k \geq 0} (-1)^k \KK(W^k;\1-\xx).
\end{eqnarray}
The terms of lowest degree on the right side of this equation result only
from summands $\kk[\zz]/\mm^{\bb_{k\ell}}$ for which the support of
$\bb_{k\ell}$ is {\em minimal}, by~(\ref{eq:hot}). In other words, only
the summands of $W^\spot$ whose associated primes $\mm^{D_L}$ have {\em
maximal} dimension $\dim(\LL)$ contribute to the sum of lowest degree
terms, of degree $|D_L| = m - \dim(\LL)$.
Let $L \subseteq \LL$ be a reduced coordinate subspace of maximal
dimension, so $\mm^{D_L}$ is an associated prime of maximal dimension.
Collecting the coefficients on the terms expressed as
$\log_\xx(\zz^{D_L})$ on the right hand side of~(\ref{eq:NW}) returns the
alternating sum $\sum_k (-1)^k \mult_L(W^k)$, by~(\ref{eq:hot}) again.
Fortunately, this alternating sum of multiplicities also equals
$\mult_L(\kk[\zz]/J)$; indeed, this is a consequence of localizing
$W^\spot$ at $\mm^{D_L}$, because length is additive on short exact
sequences of finite-length modules. We conclude that the reduced
coordinate subspace $L$ contributes $\mult_L(\kk[\zz]/J)$ to the
coefficient on $\log_\xx(\zz^{D_L})$ in~(\ref{eq:NW}), as required by
Definition~\ref{defn:geomdeg}.
\end{proof}
The wording in the last paragraph, including the phrases `terms expressed
as' and `contributes $\mult_L(\kk[\zz]/J)$ to the coefficient on' may
seem a bit odd. We had no choice, really: two distinct subspaces $L$
might have equal homogeneous logarithms $\log_\xx(\zz^{D_L})$. This
becomes especially obvious for the usual $\ZZ$-grading, in which every
subspace $L$ of codimension~$c$ has the same homogeneous logarithm.
Nonetheless, irreducible resolutions separate out the contributions from
distinct coordinate subspaces.
Proposition~\ref{prop:hilbdeg} holds for all of the modules $\Gamma$ in
Definition~\ref{defn:geomdeg}. The main reason is an easy lemma, which
also finds an application in Theorem~\ref{thm:additive}.
\begin{lemma} \label{lemma:shift}
For any Laurent polynomial $\KK(\xx)$ and $\aa \in \ZZ^n$, the
substituted Laurent polynomials obtained from $\KK(\xx)$ and
$\xx^\aa\KK(\xx)$ have equal sums of lowest degree terms.
\end{lemma}
\begin{proof}
The power series $(\1-\xx)^\aa$ has lowest degree term~$1$.
\end{proof}
\begin{cor} \label{cor:hilbdeg}
With notation as in Definition~\ref{defn:geomdeg}, the geometric
multidegree of $\Gamma = \bigoplus_{\ell=1}^r(\kk[\zz]/J_\ell)
(\aa_\ell)$ equals the sum of lowest degree terms in
$\KK(\Gamma;\1-\xx)$. This sum is a homogeneous form (in the usual
sense) of degree $m - \dim(\Gamma)$ in~$\xx$.
\end{cor}
\begin{proof}
When $r=1$, so there is only one summand, the Hilbert numerator satisfies
$\KK(\Gamma;\xx) = \xx^{\aa_1}\KK(\kk[\zz]/J_1;\xx)$. Using
Lemma~\ref{lemma:shift}, the corollary for $r=1$ follows from
Proposition~\ref{prop:hilbdeg} and the definitions. The case $r \geq 1$
follows from the additivity of Hilbert numerators for direct sums.
\end{proof}
The previous result motivates the algebraic definition of multidegree.
\begin{defn} \label{defn:multideg}
The \bem{multidegree} of a $\ZZ^n$-graded $\kk[\zz]$-module $\Gamma$ is
the sum $\cC(\Gamma;\xx)$ of the lowest degree terms in the substituted
Hilbert numerator $\KK(\Gamma;\1-\xx)$. If $\Gamma = \kk[\zz]/I$ is the
coordinate ring of a subscheme $X \subseteq \kk^m$, then write
$[X]_{\ZZ^n} = \cC(\Gamma;\xx)$.
\end{defn}
Now we can finally show that the multidegree of $\kk[\zz]/\IN(I)$ is
independent of the term order used to calculate the initial ideal
$\IN(I)$, for any $\ZZ^n$-graded ideal~$I$. More generally, we work with
initial modules for submodules of free modules.
\begin{prop} \label{prop:multideg}
Fix a $\ZZ^n$-graded module~$\Gamma$, and let $\Gamma \cong F/K$ be an
expression of~$\Gamma$ as the quotient of a free module~$F$ with
kernel~$K$. The multidegree $\cC(\Gamma;\xx)$ equals the geometric
multidegree of $F/\IN(K)$ for any initial submodule $\IN(K)$ of~$K$. In
particular, $\cC(\Gamma;\xx)$ is a homogeneous polynomial of degree $m -
\dim(\Gamma)$.
\end{prop}
\begin{proof}
This follows from Corollary~\ref{cor:hilbdeg}, because Hilbert
series---and therefore multidegrees---are preserved under taking initial
submodules. Note that $\IN(K)$ does indeed have the form of the direct
sum in Corollary~\ref{cor:hilbdeg}.
\end{proof}
Multidegrees are truly geometric invariants, depending only on the
support of a module (i.e.\ the union of all subvarieties along which the
module has nonzero multiplicity) and the module's multiplicities along
the irreducible components.
\begin{thm} \label{thm:additive}
Fix a $\ZZ^n$-graded module~$\Gamma$, and let $X_1,\ldots,X_r$ be the
maximal-dimensional irreducible components of the support variety
of\/~$\Gamma$. Then
\begin{eqnarray*}
\cC(\Gamma;\xx) &=& \sum_{\ell=1}^r\mult_{X_\ell}(\Gamma) \cdot
[X_\ell]_{\ZZ^n}.
\end{eqnarray*}
\end{thm}
\begin{proof}
Choose a filtration $\Gamma = \Gamma_s \supset \Gamma_{s-1} \supset
\cdots \supset \Gamma_0 = 0$ in which the successive quotients
$\Gamma_e/\Gamma_{e-1}$ are $\ZZ^n$-graded shifts of quotients of
$\kk[\zz]$ by $\ZZ^n$-graded primes. The multiplicity along $X_\ell$ is
the number of times a shift of the quotient $\kk[\zz]/I(X_\ell)$ appears,
because $X_\ell$ is maximal-dimensional. The Hilbert numerator of
$\Gamma$ is the sum of the Hilbert numerators of the successive
quotients: $\KK(\Gamma;\xx) = \sum_{e=1}^s
\KK(\Gamma_e/\Gamma_{e-1};\xx)$. However, the terms
$\KK(\Gamma_e/\Gamma_{e-1};\1-\xx)$ for which
$\dim(\Gamma_e/\Gamma_{e-1}) < \dim(\Gamma)$ contribute nothing to the
multidegree of $\Gamma$, by Proposition~\ref{prop:multideg}. Finally,
all of the terms $\KK(\Gamma_e/\Gamma_{e-1};\1-\xx)$ such that
$\Gamma_e/\Gamma_{e-1}$ is a shift of $\kk[\zz]/I(X_\ell)$ contribute
$[X_\ell]_{\ZZ^n}$ to $\cC(\Gamma;\xx)$, by Lemma~\ref{lemma:shift}.
\end{proof}
We record the most easily applied special case of
Theorem~\ref{thm:additive}. It is the version of
Definition~\ref{defn:geomdeg} for arbitrary $\ZZ^n$-graded ideals.
\begin{cor} \label{cor:additive}
If the subscheme $X \subseteq \kk^m$ has maximal-dimensional irreducible
components $X_1,\ldots,X_r$, then
\begin{eqnarray*}
[X]_{\ZZ^n} &=& \sum_{\ell=1}^r\mult_{X_\ell}(X) \cdot
[X_\ell]_{\ZZ^n}.
\end{eqnarray*}
\end{cor}
Our final result on multidegrees concerns coarsening the grading.
\begin{prop} \label{prop:coarsen}
Let $T \to T'$ be a map of tori acting positively on $\kk[\zz]$, with the
action of $T$ factoring through that of~$T'$, and denote the induced map
of weight lattices by $\ZZ^{n'} \to \ZZ^n$. Any $\ZZ^n$-graded module
$\Gamma$ is also $\ZZ^{n'}$-graded, and the $\ZZ^n$-graded multidegree
$\cC(\Gamma;\xx)$ maps to the $\ZZ^{n'}$-graded multidegree
$\cC(\Gamma;\xx')$ under the natural homomorphism $\sym^c_\ZZ(\ZZ^n) \to
\sym^c_\ZZ(\ZZ^{n'})$, where $c = m - \dim(\Gamma)$.
\end{prop}
\begin{proof}
This holds when $\Gamma = \kk[\zz]/J$ is a monomial quotient, because of
Proposition~\ref{prop:hilbdeg} and the definition of
$\log_\xx(\zz^{D_L})$ appearing in the geometric multidegree. It
therefore holds for all $\ZZ^n$-graded $\Gamma$ by
Corollary~\ref{cor:hilbdeg} and Proposition~\ref{prop:multideg}.
\end{proof}
At long last, here are some examples of multidegrees; see
Section~\ref{sub:alex} for more.
\begin{example}
The usual degree of a $\ZZ$-graded ideal is a special case, as is the
bigraded degree of a doubly homogeneous ideal in two sets of variables.
These degrees are usually thought of as associated to subschemes of
projective space or a product of projective spaces.
If the support semigroup $A = \NN\cdot\{\aa_1,\ldots,\aa_m\}$ admits a
homomorphism to $\NN$ taking $\aa_i$ to $1 \in \NN$ for all~$i$, then the
induced map $\ZZ[\xx] \to \ZZ[t]$ takes the $\ZZ^n$-graded multidegree of
$\kk[\zz]/I$ to $t^{{\rm codim}\ I}$ times the ordinary $\ZZ$-graded
degree of~$I$.
\end{example}
\begin{example}
The $\ZZ^{2n}$-graded multidegree of a vexillary matrix Schubert variety
is a multi-Schur polynomial. This follows from Theorem~\ref{thm:oracle}
along with the fact that double Schubert polynomials for vexillary
permutations are multi-Schur polynomials \cite{FulDegLoc,NoteSchubPoly}.
\end{example}
\begin{example} \label{ex:mn}
In the situtation of Example~\ref{ex:grading}, the ordinary weights of
the variables are given as elements of $\sym^1_\ZZ($weight lattice$)$ in
the bottom row of the diagram. The second row contains the natural maps
on multidegree rings of~$\mn$.
$$
% | | |
\begin{array}{ccccccc}
\kk^* &\into& T &\into& T \times T^{-1} & \to & (\kk^*)^{n^2} \\
\ZZ[t] &\otno& \ZZ[\xx] &\otno& \ZZ[\xx,\yy] &\from& \ZZ[\zz] \\
t &\from& x_i &\from& x_i-y_j &\from& z_{ij}
\end{array}
$$
The similarity of this diagram to that in Example~\ref{ex:functorial}
% (the only difference in the last row is $x_i-y_j$ instead of $x_i/y_j$)
is a little bit misleading: Laurent monomials more complicated than
simple variables for exponential weights will give rise to more
complicated linear combinations for ordinary weights. Note that
\hbox{$\ZZ[\xx,\yy] \from \ZZ[\zz]$} is not surjective, since $T \times
T^{-1} \to (\kk^*)^{n^2}$ has nontrivial kernel $(\kk^*)^{\pm 1}$
consisting of the elements $(\alpha, \ldots, \alpha) \times (\alpha,
\ldots, \alpha)$. Nonetheless, the image of $\ZZ[\zz]$ inside
$\ZZ[\xx,\yy]$ still surjects onto $\ZZ[\xx]$ because $T \cap
(\kk^*)^{\pm 1}$ is trivial.%
\end{example}
\begin{remark} \label{rk:positive}
Just as Hilbert numerators can be defined for nonpositive gradings simply
by using the equivariant $K$-class, multidegrees can also be defined for
nonpositive gradings, simply by taking the terms of lowest degree in the
substituted equivariant $K$-class. Slightly odd phenomena begin to
occur, such as subspaces with multidegree zero, but only when such
subspaces have trivial equivariant $K$-class. Geometrically, it means
that the subspace can be moved off to infinity torus-equivariantly. In
any case, we shouldn't be too surprised: {\em every\/} proper subspace in
$\kk^m$ with the trivial torus action has multidegree zero (and defines
the zero element in cohomology).%
\end{remark}
\begin{remark}
It can be important to think of the multidegree as associated not to a
subscheme of~$\kk^m$, but to a multigraded quotient of $\kk[\zz]$. The
distinction arises in toric geometry, where the same multigraded ideal
can represent subschemes of various rather different toric varieties, via
their Cox homogeneous coordinate rings. In each smooth such toric
variety, the multigraded degree of $\kk[\zz]/I$ maps to the appropriately
equivariant cohomology class of the subscheme determined by~$I$.
\end{remark}
\subsection{Equivariant cohomology}\label{app:cohomology}%%%%%%%%%%%%%%%%
This appendix contains the few facts about equivariant cohomology that we
use in Section~\ref{sub:loci} and Appendix~\ref{app:basics}. Most of
this is very standard (see \cite{GuilMomMapTn,GuZara} for combinatorial
approaches, or \cite{BrionEqCohEqInt,EGequivInt} for ones closer to
algebraic geometry). The only nonstandard point comes when we want to
associate equivariant cohomology classes to closed subvarieties of a
noncompact space (since the references above require compactness). We
repeat%
%
\footnote{for the convenience of the reader, as this issue of
this journal seems to be largely unavailable}
%
the argument from \cite{Kaz97} that works in the special case we need.
The field in this appendix is $\kk = \CC$.
Given a locally compact group $G$, there exists a (left) $G$-space $EG$ that
is contractible, and on which $G$ acts freely;
the quotient is called $BG$.
To an action of $G$ on another topological space $M$,
we associate the \bem{mixing space} of Borel,
\begin{eqnarray*}
M_G &=& M \times_G EG\ \:=\ \:G \dom (M \times EG).
\end{eqnarray*}
This has a natural map to $BG$ coming from projection onto the second
factor, with fibers $M$. The \bem{$G$-equivariant cohomology} $H^*_G(M)$
is defined as the cohomology of the mixing space, and the projection $M_G
\onto BG$ makes $H^*_G(M)$ into a module over $H^*(BG) = H^*_G(pt)$.
\begin{example}
The group $\CC^*$ acts on Hilbert space by rescaling. This is
contractible, but the action is not free; once we remove the zero
vector it becomes free (and the loss of an infinite-codimensional set
does not spoil the contractibility). The quotient $B\CC^*$ is thus
$\CC\PP^\infty$, and its cohomology ring is a polynomial ring $\ZZ[x]$
in one generator of degree $2$, ``the universal first Chern class''.
More generally, if $T$ is a complex torus $(\CC^*)^n$, the space
$BT$ is a product of $\CC\PP^\infty$s, and the cohomology ring is
a polynomial ring in $n$ generators, most naturally identified with
the symmetric algebra of the weight lattice $W$ of $T$.
\end{example}
If $V \onto X$ is a vector bundle, and $G$ acts on both such that the
bundle map is $G$-equivariant, the mixing construction makes $V_G \onto
X_G$ a bundle with the same fiber. In particular, one can define the
\bem{equivariant characteristic classes} of $V$, which live in
$H^*_G(X)$, as the usual characteristic classes of~$V_G$.
These two lemmata are straightforward from the definitions:
\begin{lemma} \label{lemma:quotient}
If $G \times G'$ acts on $M$ with $G = G \times \{1\}$ acting freely,
then the natural map $H^*_{G'}(G \dom M) \to H^*_{G \times G'}(M)$ is an
isomorphism.
\end{lemma}
\begin{lemma} \label{lemma:subgroup}
If $G' \subseteq G$ and $G' \dom G$ is contractible, then the natural map
$H^*_G(M) \to H^*_{G'}(M)$ is an isomorphism.
\end{lemma}
% \begin{proof}
% There is a natural map $G' \dom (EG \times M) \stackrel\pi\to G \dom (EG
% \times M)$ with contractible fibers $G' \dom G$. Since $G' \dom EG$
% suffices a model for $BG'$, we find that $H^*_{G'}(M)$ is the cohomology
% ring of the source of $\pi$, while $H^*_G(M)$ is the cohomology ring of
% the target.
% \end{proof}
Recall the notation from the beginning of Section~\ref{app:K-theory}
regarding torus actions. Define the \bem{equivariant Chern classes} of a
vector bundle $E$ by $c_i^G(E) = c_i(E_G)$.
\begin{lemma} \label{lemma:mn}
Suppose $M$ is equivariantly contractible.
\begin{thmlist}
\item \label{mn1}
The natural map $H^*(BG) \congto H^*_G(M)$ is an isomorphism.
\item \label{mn2}
If $G = T \cong (\CC^*)^n$, then $H^*_T(M) = \ZZ[W]$ consists of
polynomials in the equivariant first Chern classes of $T$-equivariant
line bundles on~$M$.
\item \label{mn3}
If $T' \to T$ is a homomorphism of tori with weight lattices $W'$ and
$W$, then \hbox{$H^*_{T'}(M) \from H^*_T(M)$} is induced by the natural
map \hbox{$W' \from W$}.
\end{thmlist}
\end{lemma}
\begin{proof}
The mixing space of an equivariant contraction of $M$ to a point is an
ordinary contraction of $M_G$ to $BG$, proving part~\ref{mn1}. Because
of the contractions, line bundles on $M_T$ and equivariant line bundles
on $M$ are (up to isomorphism) all pulled back from $BT$ and a point,
respectively. Since pullback commutes with formation of mixing
spaces, each line bundle on $M_T$ can be expressed as $L_T$ for some
equivariant line bundle $L$ on $M$. The equivariant Chern class
$c_1^T(L)$ is the ordinary Chern class $c_1(L_T)$ by definition, so
part~\ref{mn2} is a consequence of the standard isomorphism $H^*(BT) =
\sym^\spot_\ZZ(W)$. Part~\ref{mn3} is functoriality.
\end{proof}
\paragraph{Equivariant cohomology classes of subvarieties.}
\def\cC{{\mathcal C}}
\begin{prop} \label{prop:subvariety}
Let $M$ be a smooth complex variety and $G$ an algebraic group acting
algebraically on~$M$. Assume $G$ has finitely many orbits on~$M$. Every
$G$-orbit $\OO$ of codimension~$p$ in~$M$ naturally determines a
$G$-equivariant cohomology class $[\OO]_G \in H^p_G(M)$.
\end{prop}
The word ``naturally'' in the Proposition means that given a
$G$-equivariant map $\phi: M' \to M$ of varieties satisfying the
hypotheses, the class $[\OO]_G \in H^*_G(M)$ of an orbit on $M$ pulls
back under $\phi$ to the class $[\phi^{-1}(\OO)]_G \in H^*_G(M')$ of its
preimage (if the preimage has the same codimension). The only examples
of $M$ and $G$ that we actually see in this text are
$$
M = \gln \hbox{ or } \mn \hbox{ \ with \ } G = B \times B_+,
\qquad\hbox{and}\qquad M = \mn \hbox{ \ with \ } G = (\CC^*)^{n^2}.
$$
We present here the appropriate special case of the Vassiliev--Kazarian
method for constructing the equivariant cohomology classes of the
$G$-orbit closures on~$M$ (see \cite{Kaz97}). We owe a debt of gratitute
to Rich\'ard Rim\'anyi for showing us the construction that comprises the
proof below.
\begin{proof}
Let $M^p$ be the union of all $G$-orbits in $M$ of complex codimension
exactly~$p$. The mixing space $M^p_G$ of each locally closed subvariety
$M^p$ is a topological subspace of the total space $M_G$. There is a
resulting filtration $\nothing = X_{-1} \subset X_0 \subset X_1 \subset
\cdots\ $ of $M_G$ in which the $p^\th$ piece
\begin{eqnarray*}
X_p &=& \bigcup_{j \leq p} M^j_G
\end{eqnarray*}
is the union of all orbit mixing spaces having codimension at most~$p$.
The filtration $\{X_p\}$ of $M_G$ induces a filtration
$$
\cC^\spot(M_G)\ \supset\ \cC^\spot(M_G,X_0)\ \supset\ \cdots\ \supset\
\cC^\spot(M_G,X_{p-1})\ \supset\ \cC^\spot(M_G,X_p)\ \supset\ \cdots
$$
of the singular cochain complex of $M_G$. Here, $\cC^\spot(M_G,X_p)$ is
the relative cochain complex of the pair $(M_G,X_p)$ with coefficients
in~$\ZZ$. The cohomology first quadrant spectral sequence
\begin{eqnarray} \label{eq:kaz}
\delta_r : E_r^{p,q} \to E_r^{p+r,q-r+1}
\end{eqnarray}
associated to this filtration of $\cC^\spot(M_G)$ has $E_1^{p,q} =
H^{p+q}(X_p,X_{p-1})$, because of the canonical isomorphism
$\cC^\spot(M_G,X_{p-1})/\cC^\spot(M_G,X_p) = \cC^\spot(X_p,X_{p-1})$.
Even better, we can identify the relative cohomology in the $E_1$ term as
the absolute cohomology group
\begin{eqnarray} \label{eq:E1}
E_1^{p,q} &=& H^q(M^p_G)
\end{eqnarray}
of the codimension~$p$ orbit mixing space. Indeed, each of the orbits of
$G$ on $M$ is a complex submanifold, with a resulting canonically
(co)oriented normal bundle, so the isomorphism $H^{p+q}(X_p,X_{p-1})
\cong H^q(M^p_G)$ is an immediate consequence of excision followed by the
Thom isomorphism. In the case $q = 0$, the orientations on the
codimension~$p$ orbits $\OO \subseteq M$ identify generators of the
zeroth cohomology groups $H^0(\OO_G) = \ZZ$, and the $E_1$ terms on the
bottom row are direct sums
\begin{eqnarray*}
E_1^{p,0} &=& \bigoplus_{{\rm codim}(\OO)\:=\:p} H^0(\OO_G) \ \:=\
\:\bigoplus_{{\rm codim}(\OO)\:=\:p} \ZZ.
\end{eqnarray*}
All of the odd-numbered columns in the Kazarian spectral
sequence~(\ref{eq:kaz}) are zero, because the $G$-orbits on $M$ are
complex manifolds and hence even dimensional. It follows that $E_1^{p,0}
= E_2^{p,0}$ for all~$p$. Therefore, the edge homomorphisms $E_2^{p,0}
\to H^p(M_G)$ automatically induce maps $H^0(\OO_G) \to H^p(M_G)$ for
codimension~$p$ orbits $\OO$, thereby producing each cohomology class
$[\OO_G] \in H^p(M_G)$ as the image of $1 \in \ZZ = H^0(\OO_G)$. Of
course, $H^p(M_G) = H^p_G(M)$ by definition, so the class $[\OO]_G =
[\OO_G]$ is the desired equivariant cohomology class determined by the
orbit~$\OO$.
All of the constructions used in defining the $G$-equivariant cohomology
class $[\OO]_G \in H^*_G(M)$ of an orbit $\OO$ are functorial in~$M$
and~$\OO$, proving the second claim of the Proposition.
\end{proof}
The above proof demonstrates how exceedingly useful it is to be dealing
with matrix Schubert varieties and zero sets of monomial ideals, which
are orbit closures for Borel group and torus actions with finitely many
orbits. Nonetheless, our main applications, in Section~\ref{sub:schub}
and Appendix~\ref{app:basics}, require matrix Schubert varieties and
coordinate subspace arrangements to define equivariant cohomology classes
for proper subgroups of these groups. This is not a problem:
\begin{cor} \label{cor:subvariety}
Assume the hypotheses of Proposition~\ref{prop:subvariety}, and that $G'
\subseteq G$ is an algebraic subgroup.
\begin{thmlist}
\item
Every $G$-orbit $\OO$ of codimension~$p$ in~$M$ naturally determines a
$G'$-equivariant cohomology class $[\OO]_{G'} \in H^p_{G'}(M)$.
\item
Given a $G$-equivariant map $\phi: M' \to M$, the class $[\OO]_{G'} \in
H^*_{G'}(M)$ of an orbit on $M$ pulls back under $\phi$ to the class
$[\phi^{-1}(\OO)]_{G'} \in H^*_{G'}(M')$ of its preimage.
\end{thmlist}
\end{cor}
\begin{proof}
Once we've defined the class $[\OO]_G \in H^*_G(M)$, the rest comes for
free, under the natural map $H^*_G(M) \to H^*_{G'}(M)$.
\end{proof}
\subsection{Flag manifolds}\label{app:basics}%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Having supplanted the topology of the flag manifold with multigraded
commutative algebra in the main body of the exposition, we would like now
to connect back to the topological language. The material in this
appendix actually formed the basis for our original proof of
Theorem~\ref{thm:gb} over $\kk = \CC$, and therefore of
Theorem~\ref{thm:positive}, before we discovered the technology of
multidegrees.
% that allowed our exposition to remain entirely self-contained.
% We liked this material too much to leave it out completely.
The first half of this appendix also contains background material for
Sections~\ref{sub:groth} and~\ref{sub:other}.
Let $B \subset \gln$ denote the group of {\em lower} triangular matrices,
and $\fln$ the manifold of flags in $V = \CC^n$. Thus we think of $V$ as
consisting of row vectors. The group of upper triangular matrices $B_+$
has a left action on the flag manifold $\fln$ in which a matrix $b \in
B_+$ acts via right multiplication by $b^{-1}$. Identifying a
permutation $w \in S_n$ with the permutation matrix $w^T$ having a $1$~in
the $i^\th$ row and $w(i)^\th$ column, we denote by $X_w$ the Schubert
variety inside $\fln$ that is the closure of the orbit ${Bw^TB_+}$. For
instance, the smallest Schubert variety is the point $X_{w_0}$, where
$w_0$ is the \bem{long permutation} \hbox{$n\,\cdots\,2\,1$}.
We recall the identification of the class $[\OO_{X_{w_0}}] \in
K^\circ(\fln)$ in the $K$-theory of algebraic vector bundles on $\fln$
(Appendix~\ref{app:K-theory}) as a mode for introducing some more
standard
% notation.
% concepts.
objects. The ring $K^\circ(\fln)$ is generated over $\ZZ$ by
the classes $x_1, \ldots, x_n$ of the tautological line bundles $L_1,
\ldots, L_n$, which are defined as follows. To begin with, $\fln$ has
tautological subbundles $S_0, \ldots, S_n$ of the trivial bundle $\wt V =
V \otimes_\CC \OO_\fln$, the fiber of $S_i$ at a flag $F_\spot:\ 0 = F_0
\subset F_1 \subset F_2 \subset \cdots \subset F_{n-1} \subset F_n = V$
being the $i$-plane $F_i$. We set $L_i = S_i/S_{i-1}$.
It follows from the inclusion $S_i \to \wt V$ that the dual bundle
$S_i^\vee = Q_i$ is a quotient of the trivial bundle $\wt V^* = V^*
\otimes_\CC \OO_\fln$, where $V^* = \Hom_\CC(V,\CC)$. In particular, if
$e_1, \ldots, e_n$ are the standard basis row vectors in $V$, then every
dual basis vector $e_j^*$ gives rise to a section of $Q_i$, generating a
trivial line subbundle $\_i$. We therefore have maps $S_i
\otimes \_i \to \OO_\fln$ for all $i$ and $j$. The image
$e_j^*(S_i)$ is the ideal sheaf of the locus where $F_i \subseteq \ker
e_j^*$ as subspaces of $V$. Therefore, the image of the map
\begin{eqnarray} \label{noteq:S}
S &:=& \bigoplus_{i=1}^n S_i \otimes \_i
\stackrel\phi\too \OO_\fln
\end{eqnarray}
is the ideal sheaf of the locus of flags $F_\spot$ where $F_i$ is spanned
by $\{e_n, \ldots, e_{n+1-i}\}$ for all~$i$. For example, any vector $v$
spanning $F_1$ must have $(n-1)^\st$ coordinate zero because $v \in F_1
\subseteq \ker e_{n-1}^*$, and $(n-2)^\nd$ coordinate zero because $v \in
F_2 \subseteq \ker e_{n-2}^*$, and so on. The image of $\phi$ is hence
the ideal of the point $X_{w_0}$.
Now $\rank(S) = \dim(\fln)$, so the Koszul complex on $(S,\phi)$ is a
locally free resolution of the skyscraper sheaf at $X_{w_0}$. Thus
$[\OO_{X_{w_0}}] = \sum_{d \geq 0} (-1)^d [\bigwedge^d S]$. On the other
hand, the filtration of each $S_i$ by the subbundles $S_{h}$ for $h \leq
i$ has associated graded vector bundle $\gr(S_i) = \bigoplus_{h \leq i}
L_{h}$. Therefore, we have $[S_i] = \sum_{h \leq i}x_i$ in
$K^\circ(\fln)$, where again $x_i = [L_i]$. Finally, we conclude that
$[S] = \sum_{i+j \leq n} [L_i] = \sum_{i+j \leq n} x_i$, so
\begin{eqnarray*}
[\OO_{X_{w_0}}] &=& \sum_{d \geq 0}
(-1)^d[{\textstyle\bigwedge^d(\bigoplus_{i+j \leq n} L_i)}]\ \:=\ \:
\prod_{i=1}^n (1-x_i)^{n-i}\,.
\end{eqnarray*}
The polynomial $\GG_{w_0}(\xx) = \prod_{i=1}^n (1-x_i)^{n-i}$ is called
the \bem{Grothendieck polynomial for~$w_0$}. It is the ``top'' member of
a family of polynomials constructed from it in
Definition~\ref{defn:groth}.
\begin{remark} \label{rk:quirk}
The substitution $\xx \mapsto \1 - \xx$ is a change of basis accompanying
the Poincar\'e isomorphism. In general geometric terms, $c_1(L_i)$ is
the cohomology class Poincar\'e dual to the divisor~$D_i$ of the line
bundle~$L_i$, and the exact sequence $0 \to L_i^\vee \to \OO \to
\OO_{D_i} \to 0$ implies that the $K$-homology class $[\OO_{D_i}]$ equals
the $K$-cohomology class $1 - [L_i^\vee]$. Thus $\GG_w(\xx)$ writes
$[\OO_{X_w}]$ as a polynomial in the Chern characters $x_i =
e^{c_1(L_i)}$ of the line bundles~$L_i$, whereas $\GG_w(\1-\xx)$ writes
$[\OO_{X_w}]$ as polynomial (over $\QQ$, perhaps) in the expressions
\begin{eqnarray*}
1-e^{c_1(L_i)} &=& 1-e^{-c_1(L_i^\vee)}\ \:=\ \:c_1(L_i^\vee) -
\frac{c_1(L_i^\vee)^2}{2!} + \frac{c_1(L_i^\vee)^3}{3!} -
\frac{c_1(L_i^\vee)^4}{4!} + \cdots,
\end{eqnarray*}
whose lowest degree terms are the first Chern classes $c_1(L_i^\vee)$ of
the dual bundles $L_i^\vee$. Forgetting the higher degree terms here, in
Definition~\ref{defn:sqfreedeg}, and in Lemma~\ref{lemma:schubert}
%, and in Definition~\ref{defn:multideg}
amounts to taking the image in the associated graded ring of
$K_\circ(\fln)$, which is $H^*(\fln)$. See \cite[Chapter 15]{FulIT} for
details.
It is an often annoying quirk of history that we end up using the same
variable $x_i$ for both $c_1(L_i^\vee) \in H^*$ and $[L_i] \in K^\circ$.
We tolerate (and sometimes even come to appreciate) this confusing abuse
of notation because it can be helpful at times. In terms of algebra, it
reinterprets the displayed equation as: the lowest degree term in
$1-e^{-x_i}$ is just $x_i$ again.
% In any case, where it matters, we will always make it clear which we
% mean.
\end{remark}
%\subsection{The oracle}\label{sub:oracle}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the rest of this appendix, we use known statements about the relation
between Schubert polynomials and the cohomology of the flag manifold to
derive a statement about the torus-equivariant cohomology classes of
matrix Schubert varieties in~$\mn$---that is, a statement about
multidegrees.
According to the standard literature \cite{Borel53}, the cohomology
$H^*(\fln)$ with coefficients in $\ZZ$ is the quotient of the polynomial
ring $\ZZ[\xx]$ in the Chern classes $x_i = c_1(L_i^\vee)$ by the ideal
$K_n = \$ generated by the nonconstant
elementary symmetric functions. (These relations hold in $H^*(\fln)$
because the trivial rank $n$ bundle $\wt V^*$ has Chern roots $x_1,
\ldots, x_n$.) We require an\comment{how new is this, in view of others'
research?} equivariant cohomological justification for this presentation.
As in Appendix~\ref{app:K-theory}, denote by $\CC_{-\ee_i}$ the
$1$-dimensional representation of $T = (\CC^*)^n$ whose weight is the
negative of the $i^\th$ standard basis vector of the weight lattice.
\begin{lemma} \label{lemma:borel}
Let $x_i = c_1^T(\CC_{-\ee_i} \otimes \OO_\mn)$ be the torus-equivariant
Chern class. There is a natural map $\ZZ[\xx] = H^*_T(\mn) \to
H^*(\fln)$ sending $x_i \mapsto c_1(L_i^\vee)$.
\end{lemma}
\begin{proof}
We begin by justifying the following diagram.
\begin{equation} \label{eq:diagram}
\begin{array}{ccccc}
X_w
& &
& & \ol X_w
\\
\cap
& &
& & \cap
\\
\fln
&\otno&\gln
&\into&\mn
\\[5pt]
H^*(\fln)
&\congto&H^*_T(\gln)
&\otno& H^*_T(\mn)
\end{array}
\end{equation}
The isomorphism $H^*(\fln) \congto H^*_B(\gln)$ is by
Lemma~\ref{lemma:quotient} with $G = B$ and $G'$ trivial, while
$H^*_B(\gln) = H^*_T(\gln)$ by Lemma~\ref{lemma:subgroup}. Explicitly,
the isomorphism $H^*(\fln) \congto H^*_T(\gln)$ takes the first Chern
class of $L_i^\vee$ to the equivariant Chern class $c_1^T(\wt L_i^\vee)$
of its pullback to $\gln$.
The bundle $\wt L_i$ is isomorphic $T$-equivariantly (but not
$B$-equivariantly) to the line bundle on $\gln$ whose fiber over an
invertible matrix is the subspace of $V = \CC^n$ spanned by its $i^\th$
row. Moreover, the inclusion $\wt L_i \into V \times \gln$ into the
trivial bundle (with fiber $V$ having trivial $T$-action) is
$T$-equivariant.
% Indeed if $\lambda \in T$, and $Z \in \gln$ has image $[Z]$ in $\fln$,
% the isomorphism $(\wt L_i)_Z \to (\wt L_i)_{\lambda Z}$ is the identity
% map $(L_i)_{[Z]} = (L_i)_{[\lambda Z]}$.
The dual bundle $\wt L_i^\vee$ is therefore generated by its sections
$z_{i1}, \ldots, z_{in}$ of {\em weight zero}. In other words, the
canonical map $V \otimes_\CC \CC[\zz] \to \CC[\zz](\ee_i)$, whose image
is the ideal $\(\ee_i)$, becomes a surjection $V
\otimes \OO_{\gln} \onto \wt L_i^\vee$ when restricted to $\gln$
(concretely, this is because the generic $n \times n$ determinant is in
the ideal generated by any single row). In particular, $\wt L_i^\vee$ is
isomorphic to the restriction of $\CC_{-\ee_i} \otimes \OO_\mn$ to
$\gln$.
Applying the mixing construction for $T$ to these line bundles, we find
that $x_i$ pulls back to $c_1^T(\wt L_i^\vee)$ under the inclusion $\gln
\into \mn$. The second part of Lemma~\ref{lemma:mn} implies that
$\ZZ[\xx] = H^*_T(\mn)$, and completes the proof.
\end{proof}
Ehresmann \cite{Ehr} showed that the Schubert classes $[X_w] \in
H^*(\fln)$ are a $\ZZ$-basis. In the equivariant setting on $\mn$, where
the classes are uniquely defined as polynomials, the Schubert classes
arise from the matrix Schubert varieties.
\begin{lemma} \label{lemma:oracle}
The polynomial $[\ol X_w]_T \in H^*_T(\mn) = \ZZ[\xx]$ represents the
Schubert class $[X_w] \in H^*(\fln)$ under the map $\ZZ[\xx] \to
H^*(\fln)$ of Lemma~\ref{lemma:borel}.
\end{lemma}
\begin{proof}
Given that $T$-stable subvarieties of $\mn$ and $\gln$ determine
$T$-equivariant cohomology classes (Proposition~\ref{prop:subvariety}),
it is clear from (\ref{eq:diagram}) that $[\ol X_w]_T \in H^*_T(\mn)$
maps to $[\ol X_w \cap \gln]_T \in H^*_T(\gln)$, which equals $[X_w] \in
H^*(\fln)$.
\end{proof}
The insight of Lascoux and Sch\"utzenberger \cite{LSpolySchub}, based on
work of Bernstein-Gel$'$fand-Gel$'$fand \cite{BGG} and Demazure
\cite{Dem}, was to represent the classes $[X_w] \in \ZZ[\xx]/K_n$ of
Schubert varieties independently of $n$, using Schubert polynomials. To
make a precise statement, let $\flN$ be the manifold of flags in $\CC^N$
for $N \geq n$, so $B$ is understood to consist of $N \times N$ lower
triangular matrices. Let $X_{w,N} \subseteq \flN$ and $\SS_{w,N}$ be the
Schubert variety and polynomial for the permutation $w \in S_n$
considered as an element of $S_N$ that fixes $n+1, \ldots, N$.
\begin{prop}[\cite{LSpolySchub}]\label{prop:stability}
$\!\SS_w(\xx)$ is the unique polynomial representing the class $[X_{w,N}]
\in H^*(\flN)$ for all $N\geq n$.
\end{prop}
\begin{proof}
The Appendix to \cite{NoteSchubPoly} is a self-contained proof whose only
prerequisite other than the elementary algebra of divided differences
(especially \cite[(4.5)]{NoteSchubPoly}) is a formula of Monk
\cite{Monk}. Note that $X_{w_0w}$ in \cite{NoteSchubPoly} corresponds to
$X_w$ here.
\end{proof}
Here is a geometric explanation for the naturality of Schubert
polynomials.
\begin{thm} \label{notthm:oracle}
The class $[\ol X_w]_T \in H^*_T(\mn)$ is the Schubert polynomial
$\SS_w(\xx)$.
\end{thm}
\begin{proof}
We can deduce the result from Proposition~\ref{prop:stability} and
Lemma~\ref{lemma:oracle} as soon as we show why $[\ol X_w]_T$ is
independent of $n$. But $\ol X_{w,N} \subseteq \mN$ is the preimage of
$\ol X_w$ under the projection $\mN \to \mn$ forgetting the last $N-n$
rows and columns. Hence $[\ol X_{w,N}]_T$ is the image of $[\ol X_w]_T$
under the pullback map $H^*_T(\mn) \to H^*_T(\mN)$. This pullback map is
the inclusion $\ZZ[x_1, \ldots, x_n] \into \ZZ[x_1, \ldots, x_N]$.
\end{proof}
Exactly the same proof gives a stronger statement: the class $[\ol X_w]_T
\in H^*_{T\times T}(\mn)$ is the {\em double} Schubert polynomial.
\begin{remark}
It is the projection $\mN \to \mn$ that is crucial here but unavailable
when dealing with $\fln$ or even $\gln$: the result of removing the last
row and column of an invertible matrix need not be invertible.%
\end{remark}
\begin{cor} \label{cor:oracle}
The $\ZZ$-graded degree of $\ol X_w$ is $\deg(I(\ol X_w)) = \SS_w(1,
\ldots, 1)$.
\end{cor}
\begin{proof}
By Example~\ref{ex:mn}, substituting $x_i = t$ in $\SS_w$ for all $i$
yields the $\CC^*$-equivariant cohomology class $\SS_w(t, \ldots, t) =
\SS_w(1,\ldots,1)t^{\length(w)}$ for $\ol X_w$. By an argument
analogous to that of Lemma~\ref{lemma:oracle}, we have
$H^*_{\CC^*}(\mn) \onto H^*_{\CC^*}(\mn \minus \{\0\}) =
H^*(\PP^{n^2-1}) = \ZZ[t]/t^{n^2}$. In general, the degree of an
$\ell$-dimensional variety in projective space is the coefficient on
$t^\ell$ in its cohomology class. Here, we take that variety to be the
projective variety associated to the $\ZZ$-graded ideal $I(\ol X_w)$.
\end{proof}
Corollary~\ref{cor:oracle} can enter into a proof of Theorem~\ref{thm:gb}
as follows. Calculate the multidegree of $\LL_w = $ zero set of~$J_w$ as
in Section~\ref{sub:gb}, using the combinatorial machinery in
Section~\ref{sec:gb}. This yields $[\LL_w]_{\ZZ^n} = \SS_w(\xx)$.
% using Lemma~\ref{lemma:schubert}, because the Hilbert numerator of
% $\kk[\zz]/J_w$ is the Grothendieck polynomial by
% Corollary~\ref{cor:induction}. (In this case, we know that the
% $T$-equivariant cohomology class $[\LL_w]_T$ resulting from
% Corollary~\ref{cor:subvariety} with $G = (\CC^*)^{n^2}$ equals the
% $\ZZ^n$-graded multidegree $[\LL_w]_{\ZZ^n}$, by the direct computation
% of both for coordinate subspaces, plus the additivity of cohomology and
% multidegrees on irreducible components.)
Coarsening to the $\ZZ$-grading by Proposition~\ref{prop:coarsen} yields
$\deg(J_w) = \SS_w(1,\ldots,1)$. Therefore, we can apply the
$\ZZ$-graded version of Lemma~\ref{lemma:IN} and complete the proof of
Theorem~\ref{thm:gb} as in Section~\ref{sub:gb}.
Observe that this argument does {\em not} directly show the equality
$[\ol X_w]_{\ZZ^n} = [\ol X_w]_T = \SS_w(\xx)$ in
Theorem~\ref{thm:positive} between the multidegree and the
$T$-equivariant cohomology class. However, once Theorem~\ref{thm:gb} is
known, we can still conclude that the $\ZZ^{2n}$-graded multidegrees
$[\ol X_w]_{\ZZ^{2n}}$ and $[\LL_w]_{\ZZ^{2n}}$ are equal, by
Proposition~\ref{prop:multideg}.
% \begin{cor} \label{cor:oracle}
% The multidegree of $\ol X_w$ satisfies $[\ol X_w]_{\ZZ^n} = [\ol X_w]_T
% = \SS_w(\xx)$.
% \end{cor}
%\end{section}{Appendix}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\addcontentsline{toc}{section}{\numberline{}References}
\footnotesize
%\bibliographystyle{amsalpha}
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\vbox{\footnotesize \baselineskip 10pt
\bigskip\noindent
Department of Mathematics, University of California, Berkeley, CA
94720
e-mail: {\tt allenk@math.berkeley.edu}
\smallskip\noindent
Applied Mathematics, 2-363A, Massachusetts Institute of Technology, 77
Massachusetts Avenue,
Cambridge, MA 02139 \qquad email: {\tt ezra@math.mit.edu}}
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