%kquiv.tex
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%%%%%%% Alternating formulae for K-theoretic quiver polynomials %%%%%%%
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%%%%%%% Ezra Miller %%%%%%%
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\begin{document}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mbox{}\vspace{2.48ex}
\title{Alternating formulas for \K-theoretic quiver polynomials}
\author{Ezra Miller}
\thanks{The author was partly supported by the National Science
Foundation, grant DMS-0304789}
\address{University of Minnesota\\ Minneapolis, Minnesota}
\email{ezra@math.umn.edu}
\date{10 June 2004}
\begin{abstract}
\noindent
The main theorem here is the \K-theoretic analogue of the
cohomological `stable double component formula' for quiver polynomials
in \cite{quivers}. This \K-theoretic version is still in terms of
lacing diagrams, but nonminimal diagrams contribute terms of higher
degree. The motivating consequence is a conjecture of Buch on the
sign-alternation of the coefficients appearing in his expansion of
quiver \K-polynomials in terms of stable Grothendieck polynomials for
partitions \cite{Buch02}.
%\vskip 1ex
%\noindent
%{{\it AMS Classification:} ; }
\end{abstract}
\maketitle
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%\eject
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\section*{Introduction}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
The study of combinatorial formulas for the degeneracy loci of quivers
of vector bundles with arbitrary ranks was initiated by Buch and
Fulton~\cite{BF99}. In their paper they studied the case of a
sequence $E_0 \to E_1 \to \cdots \to E_n$ of vector bundles over a
fixed base. Their Main Theorem implies that given an integer array
$\rr = (r_{ij})_{i \leq j}$, the cohomology class of the
locus~$\Omega_\rr$ in the base where $E_i \to E_j$ has rank at
most~$r_{ij}$ can be expressed, under suitably general conditions, as
an integer sum of products of Schur polynomials evaluated on the Chern
classes of the bundles~$E_i$. After giving an explicit
algorithmic---but nonpositive---expression for the {\em quiver
coefficients}\/ appearing in the sum, they also conjectured a positive
combinatorial formula for them. This conjecture was proved in
\cite{quivers} (for a natural choice of an array of rectangular
tableaux) by way of three other positive combinatorial formulas for
the {\em quiver polynomials}.
The cohomological ideas of \cite{BF99} were extended to \K-theory in
\cite{Buch02}, where the class~$\KQ_\rr$ of the structure sheaf of the
degeneracy locus~$\Omega_\rr$ is expressed as an integer sum of
products of stable double Grothendieck polynomials for Grassmannian
permutations. Buch proved an algorithmic combinatorial formula for
the coefficients in this expansion of~$\KQ_\rr$
\cite[Theorem~4.1]{Buch02}, and conjectured that the signs of these
coefficients alternate in a simple manner
\cite[Conjecture~4.2]{Buch02}.
The purpose of this paper is to prove Buch's conjecture
(Theorem~\ref{t:alt}) by way of combinatorial formulas for~$\KQ_\rr$.
The starting point is a formula from \cite{quivers} that expresses a
`doubled' version of the Laurent polynomial~$\KQ_\rr$ as a ratio of
two Grothendieck polynomials (Definition~\ref{d:ratio}).
Consequently, the arguments and results in this paper do not require
any geometry of degeneracy loci or \K-theory. For an introduction to
those geometric perspectives, see \cite{Buch02}.
The main result here is Theorem~\ref{t:formula}, which gives a
\K-theoretic extension of the {\em stable double component formula}\/
that appeared in \cite[Theorem~6.20]{quivers}. That formula from
\cite{quivers} was cohomological, and was stated as a sum over {\em
lacing diagrams}\/ (graphical representations of sequences of partial
permutations) that are {\em minimal}. In Theorem~\ref{t:formula},
nonminimal diagrams contribute terms of higher degree. The word
`stable' refers to the limit in Theorem~\ref{t:formula} obtained by
adding a large constant~$m$ to all of the ranks~$r_{ij}$. When
specialized to the ordinary (non-doubled) Laurent
polynomial~$\KQ_\rr$, Theorem~\ref{t:formula} holds without taking
limits, and Buch's conjecture follows using a sign-alternation theorem
of Lascoux \cite[Theorem~4]{Las01} (see Section~\ref{sec:alt}).
The proof of Theorem~\ref{t:formula} generalizes a procedure suggested
by \cite{quivers} (see Remark~6.21 there), and carried out in
\cite{Yong03}, for constructing pipe dreams associated to given lacing
diagrams. This technique is combined with those developed in
\cite{subword} for dealing with nonreduced subwords of reduced
expressions for permutations. The \K-theory analogue of a formula
\cite[Theorem~5.5]{quivers} for quiver polynomials in terms of the
{\em pipe dreams}\/ of Fomin and Kirillov \cite{FK96} enters along
the~way~(Theorem~\ref{t:FK}).
Buch \cite{BuchAltSign} independently arrived at the main results and
definitions here (and more) by applying general techniques of Fomin
and Kirillov \cite{FK96,FK94}. A~special case of the sign conjecture
and \K-component formula already appeared in~\mbox{\cite{BKTY03}}.
\subsection*{Organization}
A notion of double quiver \K-polynomial is identified via a ratio
formula in Section~\ref{sec:quiver}, in analogy with the way
(cohomological) double quiver polynomials arise in \cite{quivers}.
The `pipe formula' for quiver \K-polynomials is proved in
Section~\ref{sec:pipe}, after background on nonreduced pipe dreams and
Demazure products. The condition on nonminimal lacing diagrams that
turns out to make them occur with sign $\pm 1$ in the \K-component
formula is defined in Section~\ref{sec:lacing}. Rank stability of
these nonminimal lacing diagrams, proved in
Section~\ref{sec:stability}, leads to the the stable \K-component
formula in Section~\ref{sec:formula}, after reviewing basics regarding
Grothendieck polynomials and their stable limits. Finally, Buch's
sign alternation conjecture is derived in Section~\ref{sec:alt}.
%\addcontentsline{toc}{subsection}{\numberline{}Acknowledgements}
%\bigskip
%\noindent
%\textbf{Acknowledgements.}
%...
%\flushbottom
%end{section}{Introduction}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Double quiver \K-polynomials}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:quiver}
A $\kl$ {\em partial permutation}\/ is a $\kl$ matrix~$w$ whose
entries are either $0$ or~$1$, with at most one nonzero entry in each
row or column. Each such matrix~$w$ can be completed to a permutation
matrix---that is, with exactly one~$1$ in each row and
column---having~$w$ as its upper-left $\kl$ corner. Viewing
permutations as lying in the union $S_\infty = \bigcup_m S_m$ of all
symmetric groups~$S_m$, there is a unique completion~$\wt w$ of~$w$
that has minimal length $\length(\wt w)$. For any partial
permutation~$w$, we write $q = w(p)$ if the entry~$w_{pq}$ at
$(p,q)$---that is, in row~$p$ and column~$q$---equals~$1$. If~$v$ is
a permutation matrix, then the assignment $p \mapsto v(p)$ defines a
permutation in~$S_\infty$.
Let $\zz = z_1,z_2,\ldots$ and $\dot\zz = \dz_1,\dz_2,\ldots$ be
alphabets. Writing a given polynomial~$f$ in these two alphabets over
the integers~$\ZZ$ as a polynomial in $z_i$ and~$z_{i+1}$ with
coefficients that are polynomials in the other variables, the {\em
$i^\th$ Demazure operator}\/ $\dem i$ sends~$f$~to
\begin{eqnarray*}
\dem if &=&
\frac{z_{i+1}f(z_i,z_{i+1})-z_if(z_{i+1},z_i)}{z_{i+1}-z_i}.
\end{eqnarray*}
Let $w_0^m$ be the permutation of maximal length in~$S_m$, and write
$s_i \in S_\infty$ for the transposition switching $i$
and~\mbox{$i+1$}. Following \cite{LS82}, the {\em double Grothendieck
polynomial}\/ for a permutation $v \in S_m$ is defined from the
``top'' double Grothendieck polynomial $\GG_{w_0^m}(\zz/\dot\zz) =
\prod_{i+j \leq m}(1-z_i/\dz_j)$ by the recursion
\begin{eqnarray*}
\GG_{vs_i}(\zz/\dot\zz) &=& \dem i \GG_v(\zz/\dot\zz)
\end{eqnarray*}
whenever $vs_i$ is lower in Bruhat order than~$v$. This definition is
independent of the choice of~$m$ \cite{LS82}. If~$w$ is a partial
permutation, then set $\GG_w(\zz/\dot\zz) = \GG_{\wt w}(\zz/\dot\zz)$.
The notation here
% ~$\GG_w(\zz/\dot\zz)$ agrees with \cite{grobGeom}, but
differs from \cite{FK94}, where their polynomial $\mathfrak
L\begin{array}{@{}l@{}}\\[-4ex]\scriptstyle (-1)\\[-1.5ex]
\scriptstyle\, w\\[-1ex]\end{array}(y,x)$ is obtained from
$\GG_w(\zz/\dot\zz)$ by replacing $z_i$ with $1-x_i$ and $\dz_i^{-1}$
with $1-y_i$.
% One can think of the $\zz$ variables as labeling the rows of~$w$,
% and $\dot\zz$ as lableling the columns.
The permutation matrices~$v$ of central importance here are those
defined as follows. Fix a positive integer~$d$ and an expression $d =
\sum_{j=0}^n r_j$ of~$d$ as a sum of $n+1$ ``ranks''~$r_j$. Endow
each $d \times d$ permutation matrix~$v$ with a block decomposition in
which the $j^\th$ block row from the top has height~$r_j$, and the
$i^\th$ block column {\em from the right}\/ has width~$r_i$. Thus
each $d \times d$ permutation matrix~$v$ is composed of $(n+1)^2$
blocks~$B_{ij}$, each of size $r_j \times r_i$ and lying at the
intersection of block column~$i$ and block row~$j$. The matrix~$v$ is
a {\em Zelevinsky permutation}\/ if $B_{ij}$ has all zero entries
whenever $i \geq j+2$, and the nonzero entries of~$v$ proceed from
northwest to southeast within every block row and within every block
column (so $v$ has no $1$~entry that is northeast of another within
the same block row or block column).
For each Zelevinsky permutation~$v$, there is a {\em rank array}\/
$\rr = (\rij)_{i \leq j}$ such that $v$ equals the Zelevinsky
permutation~$v(\rr)$ associated to~$\rr$ as in the original
construction from \cite[Proposition~1.6]{quivers}. Indeed, it can be
checked that the following suffices: define $\rij$ to be the number of
nonzero entries of~$v$ in the union of all blocks~$B_{pq}$ for which
$i \geq p$ and $j \leq q$ (that is, blocks~$B_{pq}$ weakly southeast
of~$B_{ij}$).
% Since $\rr$ uniquely determines~$v$ by
% \cite[Proposition~1.6]{quivers}, the notation $v = v(\rr)$ makes
% sense.
In particular, \mbox{$r_{ii} = r_i$},
% for all~$i$,
and the number of nonzero entries in the block $B_{j+1,j}$ of~$v$
is~$r_{j,j+1}$. Pictures and examples can be found in
\cite[Section~1.2]{quivers}, but see also Example~\ref{ex:FK},~below.
Double Grothendieck polynomials for Zelevinsky permutations $v(\rr)$
are naturally written as $\GG_{v(\rr)}(\xx/\oyy)$, using two alphabets
$\zz = \xx$ and $\dot\zz = \oyy$ each of which is an ordered sequence
of $n+1$ alphabets of sizes $r_0,\ldots,r_n$ and $r_n,\ldots,r_0$,
respectively:
\begin{eqnarray*}% \label{xx}
\xx = \xx^0,\ldots,\xx^n &\text{and}& \oyy = \yy^n,\ldots,\yy^0,
\\% \label{xxj}
\makebox[0ex][r]{where\quad} \xx^j = x^j_1,\ldots,x^j_{r_{\!j}}
&\text{and}& \yy^j = y^j_1,\ldots,y^j_{r_{\!j}}.
\end{eqnarray*}
It is convenient to think of the $\xx$ variables as labeling the rows
of the $d \times d$ grid (from top to bottom, in the above ordering on
the $\xx$ variables), while the~$\oyy$ variables label its columns
(from left to right, in the above ordering on the $\oyy$ variables).
See \cite[Section~2.2]{quivers} for pictures and examples. Most
partial permutations~$w$ that occur in the sequel will have size
$r_{j-1} \times r_{\!j}$ for some~$j \in \{1,\ldots,n\}$; in that case
we usually consider~$\GG_w(\xx^{j-1}/\yy^j)$, so $\zz = \xx^{j-1}$ and
$\dot\zz = \yy^j$.
Among all $d \times d$ Zelevinsky permutations with block
decompositions determined by $d = \sum_{j=0}^n r_j$, there is a unique
one $v(\homv)$ whose rank array $\rr(\homv)$ is maximal, in the sense
that $\rij(\homv) \geq \rij$ for all other $d \times d$\/
Zelevinsky permutations~$v(\rr)$.
\begin{defn} \label{d:ratio}
The \bem{double quiver \K-polynomial} is the ratio
\begin{eqnarray*}
\KQ_\rr(\xx/\oyy) &=& \frac{\GG_{v(\rr)}(\xx/\oyy)}
{\GG_{v(\homv)}(\xx/\oyy)}
\end{eqnarray*}
of double Grothendieck polynomials for $v(\rr)$ and~$v(\homv)$.%
\end{defn}
\noindent
The ``ordinary'' specialization of the polynomial $\KQ_\rr(\xx/\oyy)$
appears in the \K-theoretic ratio formula \cite[Theorem~2.7]{quivers}.
It will follow from Theorem~\ref{t:FK}, below, that
$\GG_{v(\homv)}(\xx/\oyy)$ divides $\GG_{v(\rr)}(\xx/\oyy)$, so the
right hand side of Definition~\ref{d:ratio} is actually a (Laurent)
polynomial rather than simply a rational function.
%end{section}{Double quiver \K-polynomials}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Nonreduced pipe dreams}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:pipe}
A $\kl$ {\em pipe dream}\/ is a subset of the \mbox{$\kl$} grid,
identified as the set of crosses in a tiling of the $\kl$ grid by {\em
crosses}\/ $\textcross$ and {\em elbow joints}\/ $\textelbow$, as in
the following diagrams:\vspace{1.5ex}
\begin{tinygraph}
\begin{dream}{5}
\hline
\* & & \* & \* & \\\hline
& \* & & & \* \\\hline
& \* & \* & & \\\hline
& & & & \* \\\hline
& \* & & & \\\hline
\end{dream}
\normal{\quad=\quad}
\begin{pipedream}{5}
\phperm&\phperm&\phperm&\phperm&\phperm\\[-4.5ex]
\+ & \jr & \+ & \+ & \jr \\
\jr & \+ & \jr & \jr & \+ \\
\jr & \+ & \+ & \jr & \jr \\
\jr & \jr & \jr & \jr & \+ \\
\jr & \+ & \jr & \jr & \jr \\
\end{pipedream}
\ \qquad\qquad
\begin{dream}{5}
\hline
\* & \* & \* & \* & \\\hline
\* & \* & \* & & \\\hline
\* & \* & & & \\\hline
\* & & & & \\\hline
& & & & \\\hline
\end{dream}
\normal{\quad=\quad}
\begin{pipedream}{5}
\phperm&\phperm&\phperm&\phperm&\phperm\\[-4.5ex]
\+ & \+ & \+ & \+ & \jr \\
\+ & \+ & \+ & \jr & \jr \\
\+ & \+ & \jr & \jr & \jr \\
\+ & \jr & \jr & \jr & \jr \\
\jr & \jr & \jr & \jr & \jr \\
\end{pipedream}
\vspace{1.5ex}
\end{tinygraph}
The square tile boundaries are omitted from the tilings forming the
newtworks of {\em pipes}\/ on right sides of these equalities. Pipe
dreams are special cases of diagrams introduced by Fomin and Kirillov
\cite{FK96}; for more background, see \cite[Section~1.4]{grobGeom}.
A pipe dream~$P$ yields a word in the Coxeter generators
$s_1,s_2,s_3,\ldots\ $ of~$S_\infty$ by reading the antidiagonal
indices of the crosses in~$P$ along rows, right to left, starting from
the top row and proceeding downward \cite{BB,FK96}. The {\em Demazure
product}\/ $\delta(P)$ is obtained (as in
\cite[Definition~3.1]{subword}) by omitting adjacent transpositions
that decrease length. More precisely, $\delta(P)$ is obtained by
multiplying the word of~$P$ using the idempotence relation $s_i^2 =
s_i$ along with the usual braid relations $s_i s_{i+1} s_i = s_{i+1}
s_i s_{i+1}$ and $s_i s_j = s_j s_i$ for $|i-j| \geq 2$. Up to signs,
this amounts to taking the product of the word of~$P$ in the
degenerate Hecke algebra \cite{FK96}. Let
\begin{eqnarray*}
\PP(w) &=& \{\hbox{pipe dreams } P \mid \delta(P) = \wt w\}
\end{eqnarray*}
for a $\kl$ partial permutation~$w$ be the set of pipe dreams whose
Demazure product is the minimal completion of~$w$ to a permutation
\mbox{$\wt w \in S_\infty$}. Every pipe dream in $\PP(w)$ fits inside
the $\kl$ rectangle, and is to be considered as a pipe dream of size
$\kl$. The subset of $\PP(w)$ consisting of {\em reduced}\/ pipe
dreams (or {\em rc-graphs}\/ \cite{BB}), where no pair of pipes
crosses more than once, is denoted by~$\rp(w)$.
Here is the observation that will make the limiting arguments in
\cite[Section~6]{quivers} for reduced pipe dreams work on nonreduced
pipe dreams (see Proposition~\ref{p:ell}, below).
\begin{lemma} \label{l:dem}
Suppose that~$P \in \PP(w)$. Then the crossing tiles in~$P$ lie in
the union of all reduced pipe dreams for~$w$.
\end{lemma}
\begin{proof}
The statement is obvious if~$P$ is reduced, so suppose otherwise.
Then some pipe dream $P' \in \PP(w)$ be can be obtained by deleting a
single crossing tile from~$P$. By induction, each crossing tile
in~$P'$ lies in some reduced pipe dream for~$w$. On the other hand, a
second pipe dream $P'' \in \PP(w)$ can be obtained from~$P$ by
deleting a different crossing tile (using \cite[Theorem~3.7]{subword},
for example). Induction shows that each crossing tile in~$P''$,
including the tile $P \minus P'$, lies in a reduced pipe
dream~for~$w$.%
\end{proof}
The {\em exponential reverse monomial}\/ associated to a $d \times d$
pipe dream~$P$ is
\begin{eqnarray*}
(\1-\tilde\xx/\tilde\yy)^P &=& \prod_{+\,\in\,P} (1 - \tilde
x_+/\tilde y_+),
\end{eqnarray*}
where the variable~$\tilde x_+$ sits at the left end of the row
containing~$\textcross$ after reversing each of the $\xx$~alphabets
% in~\eqref{xxj}
before Definition~\ref{d:ratio}, and the variable~$\tilde y_+$ sits
atop the column containing~$\textcross$ after reversing each of the
$\yy$~alphabets
% in~\eqref{xxj}
there (see Example~\ref{ex:FK}).
% (The row and column labeling in \cite[Section~2.2]{quivers} is the
% one meant on the unreversed alphabets here.)
Fix $d = r_0 + \cdots + r_n$ as in Section~\ref{sec:quiver}, and let
$D_\homv$ be the Ferrers shape of all locations strictly above the
block superantidiagonal (as in \cite[Definition~1.10]{quivers}). The
region corresponding to~$D_\homv$ in the $d \times d$ grid is filled
with crossing tiles in every reduced pipe dream for every $d \times d$
Zelevinksy permutation~$v$ (to deduce this one needs only that $v$ is
devoid of nonzero entries in that region). By Lemma~\ref{l:dem},
every nonreduced pipe dream $P \in \PP(v)$ contains~$D_\homv$, as
well. (In \cite{quivers}, the crossing tiles in $D_\homv \subseteq P$
are the~$*$~entries of~$P$, but it is necessary here to consider all
of the crosses in~$P$, including those in $D_\homv$, to make the
meaning of~$\delta(P)$ clear.)
All of the crossing tiles in every pipe dream $P \in \PP(v)$ lie above
the main antidiagonal (this holds for any permutation $v \in S_d$,
since the other crosses correspond to Coxeter generators that lie
outside of~$S_d$). Hence the ``interesting'' crossing tiles in each
pipe dream $P \in \PP(v)$ all lie in the block antidiagonal and the
block superantidiagonal. In particular, any pipe dream $P \in \PP(v)$
with no crossing tiles in its antidiagonal blocks has its
``interesting'' crosses confined to the block superantidiagonal.
These kinds of pipe dreams $P \in \PP(v(\rr))$ will be central to the
next section.
Here is the \K-theoretic analogue of the reversed version
\cite[Proposition~6.9]{quivers} of the {\em pipe formula}\/ for
cohomological quiver polynomials \cite[Theorem~5.5]{quivers}.
\begin{thm}[Pipe formula] \label{t:FK}
The double quiver \K-polynomial is the alternating~sum
\begin{eqnarray*}
\KQ_\rr(\xx/\oyy) &=& \sum_{\delta(P) = v(\rr)} (-1)^{|P| -
\length(v(\rr))}(\1-\tilde\xx/\tilde\yy)^{P \minus D_\homv}
\end{eqnarray*}
of exponential reverse monomials associated to pipe dreams $P \minus
D_\homv$ for $P \in \PP(v(\rr))$. The exponent on $-1$ is the number
crosses in~$P$ minus the length $\length(v(\rr))\!$ of~$v(\rr)$.
\end{thm}
\begin{proof}
The formula of Fomin and Kirillov \cite[Theorem~2.3 and~p.~190]{FK94}
(or see \cite[Theorem~4.1 and Corollary~5.4]{subword}) implies that
for any permutation~$w$, the Grothendieck polynomial for~$w$ can be
expressed as the alternating sum
\begin{eqnarray*}
\GG_w(\zz/\dot\zz) &=& \sum_{\delta(P)=w} (-1)^{|P|-\length(w)}
(\1-\zz/\dot\zz)^P,
\end{eqnarray*}
where $(\1-\zz/\dot\zz)^P = \prod_{+ \in P} (1-z_+/\dz_+)$ is the
product of the factors $(1-z_p/\dz_q)$ such that $P$ has a crossing
tile at~$(p,q)$. On the other hand, it follows immediately from the
isobaric divided difference recursion that $\GG_w(\zz/\dot\zz)$ is
symmetric in~$z_p$ and~$z_{p+1}$ if $w(p) < w(p+1)$---that is, if $w$
has no descent at~$p$, or equivalently, if the nonzero entry of~$w$ at
$(p,w(p))$ is northwest of $(p+1,w(p+1))$. Applying the same logic
to~$w^{-1}$, we find that $\GG_w(\zz/\dot\zz)$ is symmetric in~$\dz_q$
and~$\dz_{q+1}$ whenever $(w^{-1}(q),q)$ is northwest of
$(w^{-1}(q+1),q+1)$. Consequently, the northwest-to-southeast
progression of nonzero entries in block rows and columns implies that
the double Grothendieck polynomial for any Zelevinsky permutation is
symmetric
% ?
% \dem i(f) = f ===> f is symmetric in z_i and z_{i+1}
%
% let a = z_i and b = z_{i+1}, and g = s_i f
%
% \dem f = f <==> bf - ag = (b - a)f
% <==> (1-b)f' - (1-a)g' = (1 - b - 1 + a)f'
% <==> f' - g' + ag' - bf' = (a - b)f'
% <==> f' - g' + ag' = af'
% <==> f' - g' = a(f' - g')
% <==> f' - g' = 0
% <==> f' = g'
% <==> f = g
%
in each of its $2n+2$ alphabets. The ratio in
Definition~\ref{d:ratio} now gives the desired result via the
Fomin--Kirillov formula, after using the symmetry in the $2n+2$
alphabets to reverse each one.%
\end{proof}
\begin{example} \label{ex:FK}
Let $d = 8 = 1 + 3 + 3 + 1$ and consider the Zelevinsky permutation $v
= 52361487$ at left below. The pipe dream $P$ below has
\mbox{Demazure product $\delta(P) = v$}.
\vspace{.5ex}
\begin{tinygraph}
\raisebox{-1.5ex}{\normal{v\ =\:}}\
\begin{zel4}1331
\mc{}
&\mc{}%
&\mb{}&\mb{}&\mc{}%
&\mb{}&\mb{}&\mc{}%
&\mc{}
\\\cline{2-9}
5 &\cd & \cd&\cd&\cd & \ti&\cd&\cd & \cd
\\\cline{2-9}
2 &\cd & \ti&\cd&\cd & \cd&\cd&\cd & \cd
\\
3 &\cd & \cd&\ti&\cd & \cd&\cd&\cd & \cd
\\
6 &\cd & \cd&\cd&\cd & \cd&\ti&\cd & \cd
\\\cline{2-9}
1 &\ti & \cd&\cd&\cd & \cd&\cd&\cd & \cd
\\
4 &\cd & \cd&\cd&\ti & \cd&\cd&\cd & \cd
\\
8 &\cd & \cd&\cd&\cd & \cd&\cd&\cd & \ti
\\\cline{2-9}
7 &\cd & \cd&\cd&\cd & \cd&\cd&\ti & \cd
\\\cline{2-9}
\end{zel4}
\raisebox{-1.5ex}{\normal{\,,\qquad P\ =\:}}
\begin{zel4}1331
\mc{}
&\mc{}%
&\mb{}&\mb{}&\mc{}%
&\mb{}&\mb{}&\mc{}%
&\mc{}
\\\cline{2-9}
&\+ &\+ &\+ &\+ &\jr&\jr&\+ &\je
\\\cline{2-9}
&\+ &\jr&\jr&\jr&\jr&\jr&\je&
\\
&\+ &\jr&\jr&\jr&\+ &\je& &
\\
&\+ &\+ &\jr&\jr&\je& & &
\\\cline{2-9}
&\jr&\jr&\jr&\je& & & &
\\
&\jr&\jr&\je& & & & &
\\
&\jr&\je& & & & & &
\\\cline{2-9}
&\je& & & & & & &
\\\cline{2-9}
\end{zel4}
\quad\raisebox{-1.5ex}{\normal{\ =}}\quad
\begin{zel4}1331
\mc{}
&\mc{y^3_1\!\!}%
&\mb{y^2_3\!}&\mb{y^2_2}&\mc{y^2_1}%
&\mb{y^1_3}&\mb{y^1_2}&\mc{y^1_1}%
&\mc{y^0_1\!\!}
\\[.5ex]\cline{2-9}
\ns{x^0_1}&
\pls&\pls&\pls&\pls&\cdt&\cdt&\pls&\cdt
\\\cline{2-9}
\ns{x^1_3}&
\pls&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt
\\
\ns{x^1_2}&
\pls&\cdt&\cdt&\cdt&\pls&\cdt&\cdt&\cdt
\\
\ns{x^1_1}&
\pls&\pls&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt
\\\cline{2-9}
\ns{x^2_3}&
\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt
\\
\ns{x^2_2}&
\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt
\\
\ns{x^2_1}&
\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt
\\\cline{2-9}
\ns{x^3_1}&
\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt
\\\cline{2-9}
\end{zel4}\vspace{1ex}
\end{tinygraph}
Indeed, the transposition~$s_7$ from the cross at~$(3,5)$ is canceled
by the idempotence relation from the word associated to~$P$, so
$\delta(P) = s_7s_4s_3s_2s_1s_2s_3s_5s_4 = v$.
% 12345678
% 12354678
% 12356478
% 12536478
% 15236478
% 51236478
% 52136478
% 52316478
% 52361478
% 52361487
The exponential reverse monomial associated to $P \minus D_\homv$,
which uses the given labeling of the rows and columns by variables, is
\begin{eqnarray*}
(\1-\tilde\xx/\tilde\yy)^{P \minus D_\homv} &=&
\normal{\big(1-\frac{x^0_1}{y^1_1}\big)
\big(1-\frac{x^1_2}{y^1_3}\big) \big(1-\frac{x^1_1}{y^2_3}\big)}.
\end{eqnarray*}
The summand in Theorem~\ref{t:FK} corresponding to~$P$ contributes the
negative of this product, since $\length(v) = 9$ but $P$ has $10$
crosses.%
\end{example}
Let $m+\rr$ be the rank array obtained from~$\rr$ by adding the
nonnegative integer~$m$ to each entry of~$\rr$. Let $\xx_{m+\rr}$ be
a list of finite alphabets of sizes $m+r_0,\ldots,m+r_n$, and let the
alphabets in $\oyy{}_{\!m+\rr}$ have sizes
\mbox{$m+r_n,\ldots,m+r_0$}. This notation agrees with that in
\cite[Section~4.4]{quivers}.
% (where the rank-stability of quiver degenerations was proved).
Denote by $D_\homv(m)$ the unique reduced pipe dream for the
Zelevinsky permutation $v(m+\rr(\homv))$ in~$S_{d+m(n+1)}$ associated
to the maximal irreducible rank array.
\begin{prop} \label{p:ell}
There is a fixed integer~$\ell$, independent of~$m$, such that for
every pipe dream $P \in \PP(v(m+\rr))$ with at least one
cross~$\textcross$ in an antidiagonal block, setting the last~$\ell$
variables to~$1$ in every finite alphabet from the lists $\xx_{m+\rr}$
and\/~$\oyy{}_{\!m+\rr}$ kills the exponential reverse monomial
$(\1-\tilde\xx/\tilde\yy)^{P\minus D_\homv(m)}$.%
\end{prop}
\begin{proof}
The result is immediate from Lemma~\ref{l:dem} and
\cite[Proposition~6.10]{quivers}, the latter being the analogue for
reduced pipe dreams and `ordinary' (as opposed to exponential)
monomials.%
\end{proof}
%end{section}{Nonreduced pipe dreams}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Nonminimal lacing diagrams}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:lacing}
Suppose that $\ww = (w_1,\ldots,w_n)$ is a list of partial
permutations in which $w_j$ has size $r_{j-1} \times r_j$. The
list~$\ww$ can be identified with the (nonembedded) graph in the plane
called its {\em lacing diagram}\/ in \cite[Section~3.1]{quivers},
based on diagrams of Abeasis and Del\thinspace{}\thinspace{}Fra
\cite{AD}. The vertex set of the graph consists of $r_j$
bottom-justified dots in column~$j$ for $j = 0,\ldots,n$, with an edge
connecting the dot at height~$\alpha$ (from the bottom) in
column~$j-1$ with the dot at height~$\beta$ in column~$j$ if and only
if the entry of~$w_j$ at $(\alpha,\beta)$ is~$1$. A~{\em lace}\/ is a
connected component of a lacing diagram. For example, here is the
lacing diagram associated to a partial permutation list: {\tiny
$$
\psset{xunit=4.4ex,yunit=2.8ex}
\pspicture[.1](0,0)(3,3)
\psdots(0,-1)(0,0)(1,-1)(1,0)(1,1)(2,-1)(2,0)(2,1)(2,2)(3,-1)(3,0)(3,1)
\psline(0,-1)(1,-1)(2,0)
\psline(0,0)(1,0)
\psline(1,1)(2,-1)(3,-1)
\psline(2,1)(3,0)
\endpspicture\
\ \quad\longleftrightarrow\quad\
\left(\begin{bmatrix}
1&0&0 \\
0&1&0
\end{bmatrix}\normal{,}
\begin{bmatrix}
0&1&0&0\\
0&0&0&0\\
1&0&0&0
\end{bmatrix}\normal{,}
\begin{bmatrix}
1&0&0\\
0&0&0\\
0&1&0\\
0&0&0
\end{bmatrix}\right)
$$
}
The goal of this section is to define what it means for a rank array
to equal the Demazure product~$\delta(\ww)$ of a lacing diagram~$\ww$.
That $\delta(\ww)$ is a rank array rather than a minimal lacing
diagram is in analogy with Demazure products of lists of simple
reflections, which are permutations rather than reduced
decompositions. Usually $\delta(\ww)$ will not equal the rank array
of~$\ww$ itself. In analogy with Demazure products of {\em reduced}\/
words, however, the Demazure product of a {\em minimal}\/ lacing
diagram will equal its own rank array.
%
\begin{excise}{%
%
Given a pipe dream~$P$, as in \cite[Theorem~4.4]{subword} say that
$P$ {\em simplifies}\/ to \mbox{$D \subseteq P$} if $D$ is the
lexico\-graphically first subword of~$P$ with Demazure product
$\delta(P)$. Equivalently, denoting by $P_{\leq i}$ the length~$i$
initial string of simple reflections in~$P$, the simplification $D$
is obtained from~$P$ by omitting the~$i^\th$ reflection from~$P$ for
all~$i$ such that $\delta(P_{\leq i-1}) = \delta(P_{\leq i})$.
\begin{lemma} \label{l:simp}
Suppose that $P$ is a $\kl$ pipe dream and let $\alpha|P$ be the $k
\times (\ell + \alpha)$ pipe dream obtained by adding $\alpha$
columns of~$k$~$\textcross$ tiles to the left side of~$P$. The pipe
dream $P$ simplifies to~$D$ if and only if $\alpha|P$ simplifies
to~$\alpha|D$.
\end{lemma}
\begin{proof}
Since $P$ is reduced if and only if $\alpha|P$ is reduced, we may
assume that $P$ is not reduced. Moreover, by adding $\textcross$
tiles to~$P$ one by one (from right to left in each row and top to
bottom, as usual), it is enough to prove the lemma when $|P| = 1 +
|D|$. In this case, a single pair of pipes in~$P$ crosses twice, as
does the corresponding pair of pipes (shifted to the right
by~$\alpha$) in~$\alpha|P$. The simplifications of~$P$ and
$\alpha|P$ are obtained by deleting the southwestern crossings of
the corresponding pairs of pipes.%
\end{proof}
%
}\end{excise}%
%
\begin{defn}
Suppose $P_1,\ldots,P_n$ are pipe dreams of sizes $r_0 \times r_1,
\ldots, r_{n-1} \times r_n$, and set $d = r_0 + \cdots + r_n$. Denote
by $P(P_1,\ldots,P_n)$ the $d \times d$ pipe dream in which every
block strictly above the block superantidiagonal is filled with
crossing tiles, and the superantidiagonal $r_{j-1} \times r_j$ block
in block row~\mbox{$j-1$} is the pipe dream~$P_j$.
\end{defn}
Given a $\kl$ pipe dream~$P$, let $\oP$ be the $\kl$ pipe dream that
results after rotating~$P$ through~$180^\circ$. Also, if $m \in \NN$,
write $m+P_j$ for the $k \times (\ell+m)$ pipe dream that results by
shifting all the crosses in~$P_j$ to the right by~$m$.
%
\begin{excise}{%
%
Also, recall from \cite[Theorem~3.7]{BB} the notion of {\em top}\/
pipe dream for a partial permutation~$w$, which is the unique
reduced pipe dream in~$\rp(w)$ that has no elbow tile due north of a
crossing tile.
%
}\end{excise}%
%
\begin{prop} \label{p:dem}
Fix a lacing diagram $\ww = (w_1,\ldots,w_n)$. The Demazure product
of $P(P_1,\ldots,P_n)$ is independent of $P_1,\ldots,P_n$, as long as
$\oP_j \in \PP(w_j)$ for all $j = 1,\ldots,n$.
\end{prop}
\begin{proof}
First consider an arbitrary pipe dream~$P$. Instead of the usual word
for~$P$, consider the word in $s_1,s_2,s_3,\ldots\ $ gotten by reading
the antidiagonal indices of the crosses in~$P$ from top to bottom in
each column, starting in the right column and proceeding leftward.
The results of \cite{FK96} imply that using the idempotence and
Coxeter relations to multiply this word again yields the Demazure
product~$\delta(P)$.
We need $P(P_1,\ldots,P_n)$ and $P(P_1',\ldots,P_n')$ to have equal
Demazure products whenever $\delta(\oP_j) = \delta(\oP_j')$ for
all~$j$. It follows from the definitions that $\delta(\oP_j) =
\delta(\oP_j')$ if and only if $\delta(P_j) = \delta(P_j)$, and this
latter equality is equivalent to $\delta(m+P_j) = \delta(m+P_j')$ for
all $m \in \NN$.
Reading right to left in each row as usual, by associativity of
Demazure products we only need the corresponding block rows of $P =
P(P_1,\ldots,P_n)$ and $P' = P(P_1',\ldots,P_n')$ to have equal
Demazure products. Suppose that block row $j-1$ in the the Ferrers
shape $D_\homv$ has $m$ columns, and let $v_j$ be the Demazure product
of this block row. Applying the first paragraph to corresponding
block rows of~$P$ and~$P'$, we find that the Demazure products of
block row~\mbox{$j-1$} in~$P$ and~$P'$ are obtained by using the
idempotence and Coxeter relations to multiply $\delta(m+P_j)v_j$
and~$\delta(m+P_j')v_j$.%
\end{proof}
\begin{defn}
Fix a lacing diagram~$\ww$. If, for some (and hence, by
Proposition~\ref{p:dem}, every) sequence $P_1,\ldots,P_n$ of pipe
dreams satisfying $\oP_j \in \PP(w_j)$ for all~$j$, the Demazure
product of $P(P_1,\ldots,P_n)$ is a Zelevinsky permutation~$v(\rr)$,
then we write $\delta(\ww) = \rr$ and call the rank array~$\rr$ the
\bem{Demazure product} of the lacing diagram~$\ww$.
\end{defn}
\begin{example}
For the lacing diagram~$\ww$ below, the pipe dreams $P_j$ satisfy
$\oP_j \in \PP(w_j)$.\vspace{1ex}
\begin{tinygraph}
\begin{array}{@{}c@{}}
\psset{xunit=4.4ex,yunit=2.8ex}
\pspicture[.1](0,1.5)(3,3.5)
\psdots(0,1)(1,1)(1,2)(1,3)(2,1)(2,2)(2,3)(3,1)
\psline(0,1)(1,2)(2,1)(3,1)
\psline(1,1)(2,2)
\endpspicture
\\[4ex]
\normal\ww\
\end{array}
\qquad\ \
\begin{array}{@{}c@{}c@{}c@{}}
& & \begin{minizel}3
\cline{2-4}
&\cdt&\cdt&\pls
\\\cline{2-4}
\end{minizel}
\\[1ex]
& \begin{minizel}3
\cline{2-4}
&\cdt&\cdt&\cdt\\
&\cdt&\cdt&\cdt\\
&\pls&\cdt&\pls
\\\cline{2-4}
\end{minizel}
& \quad\ \raisebox{2ex}{\normal{P_1}}
\\[4.5ex]
\begin{minizel}1
\cline{2-2}
&\cdt\\
&\cdt\\
&\cdt
\\\cline{2-2}
\end{minizel}
& \quad\ \raisebox{2ex}{\normal{P_2}}
\\[-1ex]
\\ \quad\ \raisebox{2ex}{\normal{P_3}}
\\
\end{array}
\qquad\quad
\begin{array}{@{}c@{}}
\begin{zel4}1331
\cline{2-9}
&\pls&\pls&\pls&\pls&\cdt&\cdt&\pls&\cdt
\\\cline{2-9}
&\pls&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt
\\
&\pls&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt
\\
&\pls&\pls&\cdt&\pls&\cdt&\cdt&\cdt&\cdt
\\\cline{2-9}
&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt
\\
&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt
\\
&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt
\\\cline{2-9}
&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt&\cdt
\\\cline{2-9}
\end{zel4}
\quad\normal{\ =}\
\begin{zel4}1331
\cline{2-9}
&\+ &\+ &\+ &\+ &\jr&\jr&\+ &\je
\\\cline{2-9}
&\+ &\jr&\jr&\jr&\jr&\jr&\je&
\\
&\+ &\jr&\jr&\jr&\jr&\je& &
\\
&\+ &\+ &\jr&\+ &\je& & &
\\\cline{2-9}
&\jr&\jr&\jr&\je& & & &
\\
&\jr&\jr&\je& & & & &
\\
&\jr&\je& & & & & &
\\\cline{2-9}
&\je& & & & & & &
\\\cline{2-9}
\end{zel4}
\\
\\
\normal{P = P(P_1,P_2,P_3)}
\end{array}
\end{tinygraph}
The Demazure product~$\delta(P)$ of the pipe dream $P =
P(P_1,P_2,P_3)$ at right above equals the Zelevinsky permutation~$v$
from Example~\ref{ex:FK}. If~$\rr$ is the rank array satisfying $v =
v(\rr)$, then we conclude that $\delta(\ww) = \rr$. Observe that
although $P_1$, $P_2$, and~$P_3$ are all reduced pipe dreams, the pipe
dream~$P$ is not reduced, so
% $P \in \PP(v) \minus \rp(v)$.
% Consequently,
$\ww$ is not minimal. Thus the lacing diagram~$\ww$ will give rise to
a higher degree summand in Theorem~\ref{t:formula},~below.
\end{example}
%end{section}{Nonminimal lacing diagrams}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Rank stability of lacing diagrams}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:stability}
Next we show that lacing diagrams with Demazure product~$\rr$ are
stable, in the appropriate sense, under uniformly increasing ranks
obtained by replacing $\rr$ with~\mbox{$m+\rr$}. The results and
methods in this section rely on certain parts of \cite{quivers},
namely Proposition~5.7 (which is actually \cite[Theorem~3.7]{BB}) and
Section~5.3. That material describes the elementary combinatorics
behind the transition from~$v(\rr)$ to~\mbox{$v(1+\rr)$}
\cite[Lemma~5.11]{quivers}, and the resulting effect on reduced pipe
dreams of the form $P(P_1,\ldots,P_n)$ \cite[Propositions~5.7
and~5.15]{quivers}. To avoid excessive use of the word `block' in
what follows, we use `horizontal strip~$j$' as a synonym for
`block~row~$j$'.
\begin{lemma} \label{l:fits}
If $P(P_1,\ldots,P_n) \in \PP(v(1+\rr))$ and each $\oP_{\!j}$ is the
top pipe dream for a $(1+r_{j-1}) \times (1+r_j)$ partial
permutation~$w_j$, then all crossing tiles of $P_j$ lie in the
southwest $r_{j-1} \times r_j$ rectangle of the superantidiagonal
block in horizontal strip~\mbox{$j-1$}.
\end{lemma}
Thus the superantidiagonal block in the Lemma is supposed to have a
blank row above and a blank column to the right of the southwest
$r_{j-1} \times r_j$ rectangle in question.
\begin{proof}
No reduced pipe dream for~$v(1+\rr)$ has a crossing tile on the main
superantidiagonal, by \cite[Proposition~5.15]{quivers}.
Lemma~\ref{l:dem} implies that the same is true of~$P$. It follows
that $w_j = 1 + w_j'$ for some $r_{j-1} \times r_j$ partial
permutation~$w_j'$. Consequently, the left column of~$\oP_j$ has no
crossing tiles, and shifting all crossing tiles in~$\oP_j$ one unit to
the left results in the top pipe dream for~$w_j'$. This top pipe
dream fits inside the rectangle of size $r_{j-1} \times r_j$.%
\end{proof}
Suppose $P = P(P_1,\ldots,P_n)$ is a pipe dream in which
\begin{equation} \tag{SW}
\begin{array}{@{\ }l@{}}
\hbox{$P_j$ has size $(1+r_{j-1}) \times (1+r_j)$, but
every~$\textcross$ in~$P_j$ lies in the}
\\
\hbox{southwest $r_{j-1} \times r_j$ rectangle.}
\end{array}
\end{equation}%
Write $P_j'$ for the $r_{j-1} \times r_j$ pipe dream consisting of the
southwest rectangle of~$P_j$, and then write $P' =
P(P_1',\ldots,P_n')$. Thus $P$ has block sizes consistent with
ranks~\mbox{$1+\rr$} (so the $i^\th$ antidiagonal block is square of
size $1+r_{ii}$), while $P'$ has block sizes consistent with
ranks~$\rr$ (so the $i^\th$ antidiagonal block is square of
size~$r_{ii}$). The construction can also be reversed to create~$P$
having been given the pipe dream called~$P'$.
For a pipe dream~$P$, as in \cite[Theorem~4.4]{subword} say that $P$
{\em simplifies}\/ to \mbox{$D \subseteq P$} if $D$ is the
lexico\-graphically first subword of~$P$ with Demazure product
$\delta(P)$. Equivalently, if $P_{\leq m}$ is the length~$m$ initial
string of simple reflections in~$P$, the simplification $D$ is gotten
by omitting the~$m^\th$ reflection from~$P$ for all~$m$ such that
\mbox{$\delta(P_{\leq m-1}) = \delta(P_{\leq m})$}.
Given a reduced pipe dream~$D$, an elbow tile is {\em absorbable}\/
\cite[Section~4]{subword} if the two pipes passing through it
intersect in a crossing tile to its northeast. It follows from the
definitions that a pipe dream~$P$ simplifies to~$D$ if and only if $P$
is obtained from~$D$ by changing (at will) some of its absorbable
elbow tiles into crossing tiles.
\begin{lemma} \label{l:abs}
Suppose $D = (D_1,\ldots,D_n)$ satisfies the (SW) condition. Then $D$
is a reduced pipe dream for~$v(1+\rr)$ if and only if $D' =
(D_1',\ldots,D_n')$ is a reduced pipe dream for~$v(\rr)$. In this
case, the absorbable elbow tiles in horizontal strip~\mbox{$j-1$}
of~$D'$ are in bijection with the absorbable elbow tiles in the
southwest $r_{j-1} \times r_j$ rectangle of the superantidiagonal
block in \mbox{horizontal strip~$j-1$ of~$D$}.
\end{lemma}
\begin{proof}
The first claim is a straightforward consequence of
\cite[Proposition~5.15]{quivers}. The second claim follows because
the corresponding pairs of pipes in~$D$ and~$D'$ pass through
corresponding elbow tiles. The rest of the proof makes this statement
precise.
Given a nonzero entry of the Zelevinsky permutation~$v(1+\rr)$,
exactly one of the following three conditions must hold: (i)~the entry
lies in the northwest corner of some superantidiagonal block; (ii)~the
entry lies in the southeast corner of the whole matrix; or (iii)~there
is a corresponding nonzero entry in~$v(\rr)$. This means that the
pipes in~$D'$ are in bijection with those pipes in~$D$ corresponding
to nonzero entries of~$v(1+\rr)$ that do not satisfy (i) or~(ii).
Furthermore, it is easily checked that the pipes in~$D$ of type~(i)
or~(ii) can only intersect a superantidiagonal block in its top row or
rightmost column. Hence to say
\begin{quote}
the two pipes passing through an elbow tile in the southwest
\mbox{$r_{j-1} \times r_j$} rectangle of the superantidiagonal block
in horizontal strip~$j-1$ of~$D$ correspond to the pipes passing
through the corresponding elbow in~$D'$
\end{quote}
actually makes sense. That this claim is true follows from
\cite[Proposition~5.15]{quivers}, and it immediately proves the
lemma.%
\end{proof}
In \cite[Corollary~5.16]{quivers} it was proved that the set of lacing
diagrams obtained from reduced pipe dreams for~$v(\rr)$ is in
canonical bijection with the set of (automatically minimal) lacing
diagrams obtained from reduced pipe dreams for~$v(1+\rr)$. Here is
the extension to nonreduced pipe dreams, via nonminimal lacing
diagrams. For notation, given $m \in \NN$ and a $\kl$ partial
permutation~$w$, define the $(m+k) \times (m+\ell)$ partial
permutation $m+w$ to fix $1,\ldots,m$ and act on $\{m+1,\ldots,m+k\}$
just as $w$ acts on $\{1,\ldots,k\}$. For a list $\ww =
(w_1,\ldots,w_n)$ of partial permutations, set $m + \ww =
(m+w_1,\ldots,m+w_n)$. This notation agrees with that in
\mbox{\cite[Section~4.4]{quivers}}.
\begin{prop} \label{p:L}
For each array~$\rr$, let $L(\rr) = \{\ww \mid \delta(\ww) = \rr\}$ be
the set of lacing diagrams~$\ww$ with Demazure product\/~$\rr$. Then
$L(\rr)$ and $L(m+\rr)$ are in \mbox{canonical bijection}:
\begin{eqnarray*}
L(m+\rr) &=& \{m+\ww \mid \ww \in L(\rr)\}.
\end{eqnarray*}
\end{prop}
\begin{proof}
It suffices to prove the case $m = 1$, so suppose $\ww \in L(1+\rr)$.
Let $P = P(P_1,\ldots,P_n)$ be the pipe dream in $\PP(v(1+\rr))$ for
which each~$\oP_j$ is the top pipe dream in~$\rp(w_j)$. Then $P$
simplifies to a reduced pipe dream $D \in \rp(v(1+\rr))$. By
Lemma~\ref{l:fits} there is a corresponding pipe dream $D' \in
\rp(v(\rr))$, constructed via the procedure after Lemma~\ref{l:fits}.
On the other hand, the pipe dream~$P'$ constructed from~$P$ results by
changing back into crossing tiles those elbow tiles in~$D'$ that
correspond to the $\textcross$ tiles deleted from~$P$ to get~$D$.
Lemma~\ref{l:abs} says that $P'$ has Demazure product~$v(\rr)$.
Defining $\ww'$ by the equality $1+\ww' = \ww$, which can be done by
Lemma~\ref{l:fits}, it follows that $\ww' \in L(\rr)$.
In summary, we have constructed $P'$ from~$P$ via the intermediate
steps
$$
P \in \PP(v(1+\rr))\ \goesto\ D \in \rp(v(1+\rr))\ \goesto\ D' \in
\rp(v(\rr))\ \goesto\ P' \in \PP(v(\rr)),
$$
where the first and third steps are simplification and
``unsimplification''. Consequently, $L(1+\rr) \subseteq \{1+\ww' \mid
\ww' \in L(\rr)\}$. But the arguments justifying these steps are all
reversible, so the reverse containment holds, as well.%
\end{proof}
%end{section}{Rank stability of lacing diagrams}%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Stable double component formula}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec:formula}
The main result in this paper, namely Theorem~\ref{t:formula},
involves {\em stable double Grothen\-dieck polynomials}\/
$\hat\GG_w(\zz/\dot\zz)$ for $\kl$ partial permutations~$w$
\cite{FK94}, which we recall presently. Suppose that the argument of
a Laurent polynomial~$\GG$ is naturally a pair of alphabets
$\zz$~and~$\dot\zz$ of sizes $k$ and~$\ell$, respectively. In this
section and the next, the convention is that if $\GG(\zz/\dot\zz)$ is
written, but $\zz$ or $\dot\zz$ has {\em fewer}\/ than the required
number of letters, then the rest of the letters are assumed to
equal~$1$. For example, the notation
$\KQ_{m+\rr}(\xx_\rr/\oyy{}_{\!\rr})$ indicates that all variables in
$\xx_{m+\rr} \minus \xx_\rr$ and $\oyy{}_{\!m+\rr} \minus
\oyy{}_{\!\rr}$ (see the paragraph preceding Proposition~\ref{p:ell})
are to be set equal to~$1$.
Under this convention, let $w$ be a $\kl$ partial permutation, and
write $\GG_{m+w}(\zz_k/\dot\zz_\ell)$ for each $m \geq 0$ to mean the
Laurent polynomial $\GG_{m+w}$ applied to alphabets $\zz_k$
and~$\dot\zz_\ell$ of fixed sizes $k$ and~$\ell$. As~$m$ gets large,
these Laurent polynomials eventually stabilize, allowing the notation
$\hat\GG_w(\zz/\dot\zz) = \lim_{m \to \infty}
\GG_{m+w}(\zz_k/\dot\zz_\ell)$ for the stable double Grothendieck
polynomial.
Given a lacing diagram~$\ww$ with $r_j$ dots in column~$j$, for $j =
0,\ldots,n$ denote by
\begin{eqnarray*}
\GG_\ww(\xx/\oyy) &=& \GG_{w_1}(\xx^0/\yy^1)\cdots
\GG_{w_n}(\xx^{n-1}/\yy^n)
\end{eqnarray*}
the product of double Grothendieck polynomials taken over partial
permutations in the list $\ww = (w_1,\ldots,w_n)$. Add hats over
every~$\GG$ for the stable Grothendieck case.
Here now is the main result, the \K-theoretic analogue of the
(cohomological) component formula for stable double quiver polynomials
\cite[Theorem~6.20]{quivers}. For notation, recall the set $L(\rr) =
\{\ww \mid \delta(\ww) = \rr\}$ from Proposition~\ref{p:L}, define the
{\em length}\/ of the lacing diagram~$\ww$ to be the sum $\length(\ww)
= \sum_{i=1}^n \length(\wt w_i)$ of the lengths of the minimal
extensions of $w_1,\ldots,w_n$ to permutations, and set $d(\rr) =
\length(v(\rr)) - \length(v(\homv))$. (Thus $d(\rr)$ is the
``expected codimension'' of the locus~$\Omega_\rr$ from the
introduction, and~$d(\rr)$ can be described equivalently as
$\sum_{i
\mu(p+1)$. A~crucial property of arbitrary stable double Grothendieck
polynomials, proved in \cite[Theorem~6.13]{BuchLR}, is that every such
polynomial $\hat\GG_w(\zz/\dot\zz)$ has a unique expression
\begin{eqnarray*}% \label{mu}
\hat\GG_w(\zz/\dot\zz) &=& \sum_{{\rm Grassmannian}\ \mu}
\alpha_w^\mu\hat\GG_\mu(\zz/\dot\zz)
\end{eqnarray*}
as a sum of stable Grothendieck polynomials~$\hat\GG_\mu$ for
Grassmannian permutations. If $\ub\mu = (\mu_1,\ldots,\mu_n)$ is a
sequence of partial permutations such that the minimal completions
$\wt \mu_1,\ldots,\wt \mu_n$ are Grassmannian, then let us
call~$\ub\mu$ a {\em Grassmannian}\/ lacing~diagram.
\begin{cor} \label{c:formula}
If $\alpha_\ww^\ub\mu = \prod_{i=1}^n \alpha_{w_i}^{\mu_i}$ for lacing
diagrams\/~$\ww$ and Grassmannian~$\ub\mu$,~then
\begin{eqnarray*}
\fG_\rr(\xx/\oyy) &=&\ \sum_\ub\mu c_\ub\mu(\rr)
\hat\GG_\ub\mu(\xx/\oyy)\\
\makebox[0pt][r]{for the constants\qquad}
c_\ub\mu(\rr) &=& \sum_{\ww \in L(\rr)}
(-1)^{\length(\ww)-d(\rr)}\alpha_\ww^\ub\mu,
\end{eqnarray*}
where the first sum above is over all Grassmannian lacing
diagrams~$\ub\mu$.
\end{cor}
\begin{proof}
Expand the right hand side of Theorem~\ref{t:formula} using $\hat\GG_w
= \sum_\mu\alpha_w^\mu\hat\GG_\mu$.%
\end{proof}
Let $\fG_\rr(\xx/\oxx)$ be the specialization of the stable double
quiver \K-polynomial obtained by setting $\yy^j = \xx^j$ for $j =
0,\ldots,n$. Independently from Corollary~\ref{c:formula}, it follows
from \cite[Theorem~4.1]{Buch02} that the (ordinary) stable quiver
\K-polynomial
\begin{eqnarray*}% \label{fG}
\fG_\rr(\xx/\oxx) &=& \sum_{\ub\mu} c_\ub\mu(\rr)
\hat\GG_\ub\mu(\xx/\oxx)
\end{eqnarray*}
is a sum of products of stable double Grothendieck polynomials
$\hat\GG_{\mu_j}(\xx^{j-1}/\xx^j)$ for Grassmannian permutations~$\wt
\mu_j$, with uniquely determined integer coefficients~$c_\ub\mu(\rr)$.
That these coefficients are the same as in Corollary~\ref{c:formula}
follows from the fact that the right side
% of~\eqref{fG}
above determines the same element in the $n^\th$ tensor power of
Buch's bialgebra~$\Gamma$ \cite{BuchLR,Buch02} as does the right side
of the top formula in Corollary~\ref{c:formula}.
In addition to proving
% ~\eqref{mu},
the expansion of~$\hat\GG_w$ as a sum of terms
$\alpha_w^\mu\hat\GG_\mu$, Buch showed in \cite[Theorem~6.13]{BuchLR}
that the coefficients $\alpha_w^\mu$ can only be nonzero if
$\length(\mu) \geq \length(w)$, and he conjectured that the sign
of~$\alpha_w^\mu$ equals $(-1)^{\length(\mu)-\length(w)}$. This was
proved by Lascoux \cite[Theorem~4]{Las01} as part of his extension of
``transition'' from Schubert polynomials to Grothendieck polynomials.
Since, as shown in \cite[Section~5]{Buch02}, the coefficients
$\alpha_w^\mu$
% in~\eqref{mu}
are special cases of the coefficients~$c_\ub\mu(\rr)$, Lascoux's
result is evidence for the following more general statement that was
surmised by Buch (prior to \cite{Las01}).
\begin{thm}[{\cite[Conjecture~4.2]{Buch02}}]\label{t:alt}
The coefficients $c_\ub\mu(\rr)$ alternate in sign; that is,
\mbox{$(-1)^{\length(\ub\mu)-d(\rr)} c_\ub\mu(\rr) \geq 0$} is a
nonnegative integer.
\end{thm}
\begin{proof}
By \cite[Theorem~4]{Las01} the sign of~$\alpha_\ww^\ub\mu$ is
$(-1)^{\length(\ub\mu)-\length(\ww)}$. Thus the sign
of~$c_\ub\mu(\rr)$~is $(-1)^{\length(\ww) -
d(\rr)}(-1)^{\length(\ub\mu) - \length(\ww)} = (-1)^{\length(\ub\mu) -
d(\rr)}$, by the second formula in Corollary~\ref{c:formula}.%
\end{proof}
%end{section}{Sign alternation}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\raggedbottom
%\addcontentsline{toc}{section}{\numberline{}}
%\bibliographystyle{amsalpha}\bibliography{biblio}\end{document}
\def\cprime{$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\begin{thebibliography}{BKTY0\,}
\bibitem[AD80]{AD}
S.~Abeasis and A.~Del\thinspace{}\thinspace{}Fra, \emph{Degenerations
for the representations of a quiver of type $A_m$},
Boll. Un. Mat. Ital. Suppl. (1980) no. \textbf{2}, 157--171.
\bibitem[BB93]{BB}
Nantel Bergeron and Sara Billey, \emph{{RC}-graphs and {Schubert}
polynomials}, Experimental Math. \textbf{2} (1993), no.~4,
257--269.
\bibitem[BF99]{BF99}
Anders~Skovsted Buch and William Fulton, \emph{Chern class formulas
for quiver varieties}, Invent. Math. \textbf{135} (1999), no.~3,
665--687.
\bibitem[BFR03]{BFR03}
Anders S. Buch, L\'aszl\'o M. Feh\'er, and Rich\'ard Rim\'anyi,
\emph{Positivity of quiver coefficients through Thom polynomials},
preprint, 2003. \textsf{arXiv:math.AG/0311203}
\bibitem[BKTY03]{BKTY03}
Anders~S. Buch, Andrew Kresch, Harry Tamvakis, and Alexander Yong,
\emph{Grothendieck polynomials and quiver formulas}, 2003.
\bibitem[Buc02a]{Buch02}
Anders~S. Buch, \emph{Grothendieck classes of quiver varieties},
Duke Math. J. \textbf{115} (2002), no.~1, 75--103.
\bibitem[Buc02b]{BuchLR}
Anders~S. Buch, \emph{A Littlewood--Richardson rule for the \K-theory
of Grassmannians}, Acta Math. \textbf{189} (2002), 37--78.
\bibitem[Buc03]{BuchAltSign}
Anders~S. Buch, \emph{Alternating signs of quiver coefficients},
preprint, 2003.
\bibitem[FK94]{FK94}
Sergey Fomin and Anatol~N. Kirillov, \emph{Grothendieck polynomials
and the {Y}ang--{B}axter equation}, 1994, Proceedings of the Sixth
Conference in Formal Power Series and Algebraic Combinatorics,
DIMACS, pp.~183--190.
\bibitem[FK96]{FK96}
Sergey Fomin and Anatol~N. Kirillov, \emph{The {Y}ang-{B}axter
equation, symmetric functions, and {S}chubert polynomials}, Discrete
Math. \textbf{153} (1996), no.~1--3, 123--143, Proceedings of the
5th Conference on Formal Power Series and Algebraic Combinatorics
(Florence, 1993).
\bibitem[KM03a]{grobGeom}
Allen Knutson and Ezra Miller, \emph{{Gr\"{o}bner} geometry of
{Schubert} polynomials}, to appear in Ann.\ of Math~(2), 2003.
\bibitem[KM03b]{subword}
Allen Knutson and Ezra Miller, \emph{Subword complexes in {Coxeter}
groups}, to appear in Adv.\ in Math., 2003.
\textsf{arXiv:math.AG/0309259}
\bibitem[KMS03]{quivers}
Allen Knutson, Ezra Miller, and Mark Shimozono, \emph{Four positive
formulae for type~$A$ quiver polynomials}, preprint, 2003.
\textsf{arXiv:math.AG/0308142}
\bibitem[Las01]{Las01}
A. Lascoux, \emph{Transition on Grothendieck polynomials}, Physics and
combinatorics, 2000 (Nagoya), 164--179, World Sci. Publishing, River
Edge, NJ, 2001.
\bibitem[LS82]{LS82}
Alain Lascoux and Marcel-Paul Sch{\"u}tzenberger, \emph{Structure de
{H}opf de l'anneau de cohomologie et de l'anneau de {G}rothendieck
d'une vari\'et\'e de drapeaux}, C. R. Acad. Sci. Paris S\'er. I
Math. \textbf{295} (1982), no.~11, 629--633.
\bibitem[Yon03]{Yong03}
Alexander Yong, \emph{On combinatorics of quiver component formulas},
preprint, 2003.
\end{thebibliography}
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