FOUR POSITIVE FORMULAE FOR TYPE A QUIVER POLYNOMIALS

ALLEN KNUTSON, EZRA MILLER, AND MARK SHIMOZONO

Abstract. We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs). Three of our formulae are multiplicity-free and geometric, meaning that their summands have coefficient 1 and correspond bijectively to components of a torus-invariant scheme. The remaining (presently non-geometric) formula is a variant of the conjecture of Buch and Fulton in terms of factor sequences of Young tableaux [BF99]; our proof of it proceeds by way of a new characterization of the tableaux counted by quiver constants. All four formulae come naturally in “doubled” versions, two for double quiver polynomials, and the other two for their stable limits, the double quiver functions, where setting half the variables equal to the other half specializes to the ordinary case.

Our method begins by identifying quiver polynomials as multidegrees [BB82, Jos84, BB85, Ros89] via equivariant Chow groups [EG98]. Then we make use of Zelevinsky’s map from quiver loci to open subvarieties of Schubert varieties in partial flag manifolds [Zel85]. Interpreted in equivariant cohomology, this lets us write double quiver polynomials as ratios of double Schubert polynomials [LS82] associated to Zelevinsky permutations; this is our first formula. In the process, we provide a simple argument that Zelevinsky maps are scheme-theoretic isomorphisms (originally proved in [LM98]). Writing double Schubert polynomials in terms of pipe dreams [FK96] then provides another geometric formula for double quiver polynomials, via [KM03a]. The combinatorics of pipe dreams for Zelevinsky permutations implies an expression for limits of double quiver polynomials in terms of products of Stanley symmetric functions [Sta84]. A degeneration of quiver loci (orbit closures of GL on quiver representations) to unions of products of matrix Schubert varieties [Ful92, KM03a] identifies the summands in our Stanley function formula combinatorially, as lacing diagrams that we construct based on the strands of Abeasis and Del Fra in the representation theory of quivers [AD80]. Finally, we apply the combinatorial theory of key polynomials to pass from our lacing diagram formula to a double Schur function formula in terms of peelable tableaux [RS95a, RS98], and from there to our formula of Buch–Fulton type.

Contents

Introduction 3 Overview 3 Acknowledgments 4 The four formulae 4 Proofs via double versions and limits 11 Related notions and extensions 12 A note on the field 𝕜 13

Date: 16 January 2006. AK was partly supported by the Sloan Foundation and NSF. EM and MS were partly supported by the NSF.

Section 1. Geometry of quiver loci 14 1.1. Quiver loci and ideals 14 1.2. Rank, rectangle, and lace arrays 15 1.3. The Zelevinsky map 18 1.4. Quiver polynomials as multidegrees 21

Section 2. Double quiver polynomials 24 2.1. Double Schubert polynomials 24 2.2. Double quiver polynomials 25 2.3. Ratio formula for quiver polynomials 27

Section 3. Lacing diagrams 30 3.1. Geometry of lacing diagrams 30 3.2. Minimum length lacing diagrams 32

Section 4. Degeneration of quiver loci 34 4.1. Orbit degenerations 34 4.2. Quiver degenerations 35 4.3. Multidegrees of quiver degenerations 36 4.4. Rank stability of components 37

Section 5. Pipe dreams for Zelevinsky permutations 40 5.1. Pipe dream formula for double quiver polynomials 40 5.2. Pipes to laces 43 5.3. Rank stability of pipe dreams 44

Section 6. Component formulae 47 6.1. Double Stanley symmetric functions 49 6.2. Double quiver functions 51 6.3. All components are lacing diagram orbit closures 54 6.4. Stable double component formula 55

Section 7. Quiver constants 57 7.1. Demazure characters 57 7.2. Schubert polynomials as sums of Demazure characters 59 7.3. Quiver constants are Stanley coefficients 61 7.4. Peelable tableaux 63

Section 8. Factor sequences from peelable tableaux 65 8.1. Factor sequences 65 8.2. Zelevinsky peelable tableaux 68 8.3. Bijection to factor sequences 69 8.4. The Buch–Fulton conjecture 73 References 76 Introduction

Overview. Universal formulae for cohomology classes appear in a number of different guises. In topology, classes with attached names of Pontrjagin, Chern, and Stiefel–Whitney arise as obstructions to vector bundles having linearly independent sections. Universality of formulae for these and other more general classes can be traced, for our purposes, to the fact that they—the formulae as well as the classes—live canonically on classifying spaces, from which they are pulled back to arbitrary spaces along classifying maps.

In algebra, cohomology classes on certain kinds of varieties (projective space, for example) are called ‘degrees’, and many universal formulae show up in degree calculations for classical ideals, such as those generated by minors of fixed size in generic matrices [Gia04] (or see [Ful98, Chapter 14]).

In geometry, pioneering work of R. Thom [Tho55] associated cohomology classes to sets of critical points of generic maps between manifolds, where the differential drops rank. Using slightly different language, the set of critical points can be characterized as the degeneracy locus for the associated morphism of tangent bundles. Subsequently there have appeared numerous extensions of Thom’s notion of degeneracy locus, such as to maps between pairs of arbitrary vector bundles.

The cohomology classes Poincaré dual to degeneracy loci for (certain collections of) morphisms of complex vector bundles are expressible as polynomials in the Chern roots of the given vector bundles. It seems to be a general principle that coefficients in universal such expressions as sums of simpler polynomials always seem to be governed by combinatorial rules. This occurs, for example, in [Ful92, Buc01b, BKTY02]. However, even when it is possible to prove an explicit combinatorial formula, it has usually been unclear how the geometry of degeneracy loci reflects the combinatorics directly.

Our original motivation for this work was to close this gap, by bringing the combinatorics of universal formulae for cohomology classes of degeneracy loci into the realm of geometry, continuing the point of view set forth in [KM03a]. To do so, we reduce the degeneracy locus problem in [BF99] to the algebraic perspective mentioned above in terms of degrees of determinantal ideals, by way of the topological perspective mentioned above in terms of classifying spaces, as in [Kaz97, FR02b, KM03a]. This reduction allows us to manufacture formulae that are simultaneously geometric as well as combinatorial, by applying flat degenerations of subvarieties inside torus representations. The cohomology class remains unchanged in the degeneration and is given by summing the classes of all components of the special fiber.

The degenerations we employ result from one-parameter linear torus actions, and hence have Gröbner bases as their natural language. Our theorems on orbit closures and their combinatorics explicitly generalize results from the theory of ideals generated by minors of fixed size in generic matrices, where the prototypical result is Giambelli’s degree formula [Gia04]. The ideals that interest us here are generated by minors in products of generic matrices. It is intriguing that our combinatorial analysis of these general determinantal ideals proceeds (using the ‘Zelevinsky map’) via Schubert determinantal ideals [Ful92, KM03a], which are generated by minors of varying sizes in a single matrix of variables. Arbitrary Schubert conditions may seem, consequently, to be in some sense more general; but while this may be true, the special form taken by the Schubert rank conditions on single matrices here give rise to a substantially richer combinatorial structure. This richness is fully borne out only by combining methods based on Schubert determinantal ideals with an approach in terms of minors in products of generic matrices, which is rooted more directly in representation theory of quivers.

Our technique of orbit degeneration automatically produces positive geometric formulae that are universal, since they essentially live on classifying spaces. However, there still remains the elucidation of what combinatorics most naturally describes these formulae, or indexes its components. Well over half of our exposition in this paper is dedicated to unearthing rich and often surprising interconnections between a number of combinatorial objects, some known and some new. Combinatorics—of objects including Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs)— forms the bridge between positive geometric formulae and the explicit algebra of several families of polynomials, including Schur functions, Schubert polynomials, Stanley symmetric functions, and now quiver polynomials. Working in the broader context of double quiver polynomials and their stable versions, the double quiver functions, allows the extra flexibility required for the proof of our variant of the Buch–Fulton conjecture [BF99]. Special cases of this conjecture are proved in [Buc01b, BKTY02].

Acknowledgments. We would like to thank Anders Buch and Bill Fulton for suggesting the problem that concerns us here, and for patiently listening, with Sergey Fomin, Richard Stanley, and Alex Yong, to endless hours of us presenting potential and actual methods toward its solution. We are indebted to Frank Sottile and Chris Woodward for inviting us to their superb AMS Special Session on Modern Schubert Calculus at Northeastern University in October 2002; that meeting gave us the opportunity to present our positivity proof for the quiver constants (which consisted of Corollary 4.9 plus Remark 4.15) and our conjectural component formula, as well as to learn of [BKTY02] for the first time. We are grateful to the Banff International Research Station (BIRS), as well as the organizers of the May 2003 meeting on Algebraic Combinatorics there, for bringing the three of us together into the same room for the first time, thereby allowing us to complete the initial version of this paper. Comments that improved the exposition were provided by Anders Buch, Bill Fulton, Peter Magyar, Richárd Rimányi, and Frank Sottile. In particular, thanks are due to Anders Buch for pointing out how our main results imply that what we call double quiver functions are specializations of the power series in [Buc01b, Section 2], and coincide with the lowest degree terms of the K-theoretic series defined in [Buc02, Section 4]. Finally, Laura Matusevich suggested the term ‘lacing diagrams’ for their resemblance to tied shoes.

The four formulae. In this paper we consider formulae for cohomology classes of degeneracy loci for type A equioriented quivers of vector bundles. This simply means that we start with sequences E_{0} → E_{1} → ⋯ → E_{n} of vector bundle morphisms over a fixed base. Since we immediately reduce in Theorem 1.20 to considering the equivariant classes of universal degeneracy loci, or quiver loci (to be defined shortly), we present our exposition in that context, referring the reader to [BF99] for an introduction in the language of vector bundles. We work over an arbitrary field 𝕜.

Consider the vector space Hom of sequences V_{0} ⟶^{ϕ_{1}} V_{1} ⟶^{ϕ_{2}} ⋯ ⟶^{ϕ_{n−1}} V_{n−1} ⟶^{ϕ_{n}} V_{n} of linear transformations between vector spaces of dimensions r_{0}, …, r_{n}, each thought of as consisting of row vectors. An element ϕ ∈ Hom is called a quiver representation, which we view as a sequence of r_{i−1} × r_{i} matrices. Each matrix list ϕ determines its rank array r(ϕ) = (r_{ij}(ϕ))_{i≤j}, where r_{ij}(ϕ) for i < j equals the rank of the composite map V_{i} → V_{j}, and r_{ii} = dim(V_{i}). The data of a rank array determines a quiver locus Ω_{r}, defined as the subscheme of matrix lists ϕ ∈ Hom with rank array dominated by r entrywise: r_{ij}(ϕ) ≤ r_{ij} for all i ≤ j. Thus Ω_{r} is the zero scheme of the ideal I_{r} in the coordinate ring of Hom generated by all minors of size (1 + r_{ij}) in the product of the appropriate j − i matrices of variables, for all i < j (Definition 1.1).

We assume throughout that r = r(ϕ) for some matrix list ϕ ∈ Ω_{r}. This is a nontrivial condition equivalent to the irreducibility of Ω_{r}. Alternatively, it means that Ω_{r} is an orbit closure for the group GL = ∏_{i=0}^{n} GL(V_{i}) that acts on Hom by change of basis in each vector space V_{i}. In particular Ω_{r} is stable under the action of any torus in GL. Picking a maximal torus T, we define the quiver polynomial as the T -multidegree of Ω_{r}. We show in Proposition 1.19 that this polynomial coincides with the T -equivariant class of Ω_{r} in the Chow ring A_{T}^{∗} (Hom), or alternatively in cohomology when 𝕜 = ℂ. This Chow ring is a polynomial ring ℤ[x_{r}] over the integers in an alphabet x_{r} of size r_{0} + ⋯ + r_{n}, the union of bases x^{0}, …, x^{n} for the weight lattices of the maximal tori in GL(V_{i}). Since Ω_{r} is stable under GL and not just the torus T, its equivariant class is symmetric in each of the alphabets x^{i}. More naturally, this is precisely the statement that the quiver polynomial lies inside A_{GL}^{∗}(Hom) ⊂ A_{T}^{∗} (Hom), and in this sense does not depend on our choice of T. We express the quiver polynomial determined by the rank array r as Q_{r}(x − ^{∘}x) for reasons that become clear in the ratio formula and double versions.

Next we present our four positive combinatorial formulae for quiver polynomials, in the same order that we will prove them in the main body of the text: the ratio formula, the pipe formula, the component formula, and the tableau formula. Besides making the overview of their proofs more coherent, this choice will serve to emphasize an important point that is worth bearing in mind before seeing the statements: while each formula stands well enough on its own, this paper is not merely a catalog of four different perspectives on quiver loci. Connections among the formulae lend added insight to each, and transformations between them often form crucial parts of proofs.

Before going into any sort of detail about the formulae, let us put them briefly in perspective (cross-references can be found in the more detailed subsections to come). The ratio formula writes Q_{r} as a ratio of double Schubert polynomials. It arises geometrically from a comparison between the quiver locus Ω_{r} and a related matrix Schubert variety. The pipe formula expresses Q_{r} as a sum over combinatorial gadgets called pipe dreams. It is an easy consequence of the ratio formula, given that double Schubert polynomials expand as sums over the same gadgets. The component formula breaks Q_{r} into a sum of products of double Schubert polynomials. It arises geometrically because the quiver locus Ω_{r} degenerates to a limit scheme Ω_{r}(0) that is usually reducible; each component is a product of matrix Schubert varieties, so its multidegree is a product of double Schubert polynomials. The pipe formula shows up here to provide a combinatorial upper bound on which components can occur in Ω_{r}(0). Finally, the tableau formula recovers the decomposition of Q_{r} into sums of products of Schur polynomials from [BF99]. The positive combinatorial properties of the coefficients in this decomposition derive principally from the ratio formula, by expanding the numerator Schubert polynomial as a sum of Demazure characters. However, the Schurproduct decomposition of Q_{r} also results by expressing (certain limits of) the summands in the component formula in terms of Schur functions, and it is this transformation that provides the bridge to the quiver constants from [BF99].

For each formula, we shall give here a precise statement, although many details of definitions will be left for later. Each formula will be presented along with a pointer to its location in the text, and an example using the following n = 3 rank array r = (r_{ij}).

(∗)  r = \begin{aligned}3 & 2 & 1 & 0 & i⧸j \\ & & & 1 & 0 \\ & & 3 & 1 & 1 \\ & 3 & 2 & 1 & 2 \\ 1 & 1 & 1 & 0 & 3 \end{aligned}

We use a priori infinite alphabets a = x^{0}, b = x^{1}, c = x^{2}, and d = x^{3} in the examples, keeping in mind that for the above ranks r, the only variables actually appearing are a_{1} from the a alphabet, b_{1}, b_{2}, b_{3} from the b alphabet, c_{1}, c_{2}, c_{3} from the c alphabet, and d_{1} from the d alphabet. With these conventions the quiver polynomial is

Q_{r} = Q_{r}(x − ^{∘}x) = (b_{1} + b_{2} + b_{3} − c_{1} − c_{2} − c_{3})(a_{1} − d_{1}).

In the general setting we distinguish between the symbol x, which denotes a sequence x^{0}, …, x^{n} of infinite alphabets x^{i} = x_{1}^{i}, x_{2}^{i}, x_{3}^{i}, …, and x_{r}, which denotes a sequence x_{r}^{0}, …, x_{r}^{n} of finite alphabets, with x_{r}^{i} having cardinality r_{i}. In the notation Q_{r}(x − ^{∘}x) for quiver polynomials, only the finitely many variables x_{r} ⊂ x actually appear, even though we view Q_{r} as taking a sequence of infinite alphabets as input.

Ratio formula (Zelevinsky permutation). Associated to each rank array r is a permutation v(r) in the symmetric group S_{d} for d = r_{0} +⋯+r_{n}. This Zelevinsky permutation v(r) has a block structure determined by r. Proposition 1.6 gives a complete characterization. Given the dimensions r_{0}, …, r_{n}, there is a unique Zelevinsky permutation v(Hom) of minimal length. It is associated to the dense GL-orbit in Hom, that is, the maximal rank array, whose ij entry is the maximum possible rank for a map V_{i} → V_{j} that factors through V_{k} for i ≤ k ≤ j. The diagram of v(Hom), as defined after Proposition 1.6, is particularly simple.

The last ingredients for our first formula are the double Schubert polynomials of Lascoux and Schützenberger [LS82] (Section 2.1). These represent torus-equivariant cohomology classes of Schubert varieties in flag manifolds and are characterized geometrically as the multidegrees of matrix Schubert varieties [FR02a, KM03a]. The double Schubert polynomials 𝔖_{w}(X − Y) are indexed by permutations w ∈ S_{d} and take as input two alphabets X and Y of size d. View x_{r} as an alphabet of size d by concatenating its n + 1 alphabets, and write ^{∘}x_{r} = x_{r}^{n}, …, x_{r}^{0} for the size d alphabet obtained by block reversing x_{r}.

Theorem (Ratio formula). Q_{r}(x − ^{∘}x) = \frac{𝔖_{v(r)}(x_{r} − ^{∘}x_{r})}{𝔖_{v(Hom)}(x_{r} − x^{∘}_{r})}.

This result is proved in Theorem 2.9. It is geometric in the sense that it interprets in equivariant cohomology the Zelevinsky map (Section 1.3), which takes quiver loci Ω_{r} isomorphically to open subvarieties of Schubert varieties in partial flag manifolds. The geometry of the Zelevinsky map explains why Schubert polynomials appear. Example. The rank array r in (∗) has the associated Zelevinsky permutations

v(r) = \begin{matrix}5 & □ & □ & □ & □ & × & · & · & · \\ 2 & □ & × & · & · & · & · & · & · \\ 3 & □ & · & × & · & · & · & · & · \\ 6 & □ & · & · & □ & · & × & · & · \\ 1 & × & · & · & · & · & · & · & · \\ 4 & · & · & · & × & · & · & · & · \\ 8 & · & · & · & · & · & · & □ & × \\ 7 & · & · & · & · & · & · & × & · \end{matrix} and v(Hom) = \begin{matrix}5 & ∗ & ∗ & ∗ & ∗ & × & · & · & · \\ 2 & ∗ & × & · & · & · & · & · & · \\ 3 & ∗ & · & × & · & · & · & · & · \\ 4 & ∗ & · & · & × & · & · & · & · \\ 1 & × & · & · & · & · & · & · & · \\ 6 & · & · & · & · & · & × & · & · \\ 7 & · & · & · & · & · & · & × & · \\ 8 & · & · & · & · & · & · & · & × \end{matrix}

The numbers running down the left sides constitute one-line notation for v(r) and v(Hom); v(r) takes 1 ↦ 5, 2 ↦ 2, 3 ↦ 3, 4 ↦ 6, and so on. We have replaced all 1 entries in the permutation matrices for v(r) and v(Hom) with × entries because it is sometimes convenient (as in Section 8.1) to use integers in these arrays for other purposes. The boxes □ denote cells in the diagram of v(r), whereas the ∗ entries denote cells in the diagram of v(Hom). That v(Hom) has such a simple diagram means that the denominator in the formula

Q_{r} = \frac{𝔖_{52361487}(a, b, c, d − d, c, b, a)}{𝔖_{52341678}(a, b, c, d − d, c, b, a)}

equals the product (a_{1} − c_{3})(a_{1} − c_{2})(a_{1} − c_{1})(a_{1} − d_{1})(b_{1} − d_{1})(b_{2} − d_{1})(b_{3} − d_{1}) of linear factors corresponding to the locations of the ∗ entries.

Pipe formula. One of many elementary ways to see that the denominator in the ratio formula is always the corresponding product of linear factors, and divides the numerator, is through our next combinatorial formula. It is in fact little more than an application of the Fomin–Kirillov double version [FK96] of the Billey–Jockusch–Stanley formula for Schubert polynomials [BJS93, FS94].

For a permutation v ∈ S_{d}, denote by RP(v) its set of reduced pipe dreams (also known as planar histories or rc-graphs). These are certain fillings the d × d grid with square tiles of the form or [figure] , in which two pipes either cross or avoid each other (Section 5.1). For example, the permutation v(Hom) has only one reduced pipe dream D_{Hom}, with crossing tiles in the diagram of v(Hom) and elbow tiles [figure] elsewhere. For Zelevinsky permutations v(r), we have D ⊇ D_{Hom} whenever D ∈ RP(v(r)) is a reduced pipe dream. We identify pipe dreams with their sets of crossing tiles as subsets of the d × d grid.

Label the rows of the grid from top to bottom with the ordered alphabet x_{r}, and label the columns from left to right with ^{∘}x_{r}. Given a pipe dream D, define (x_{r} − ^{∘}x_{r})^{D} as the product of linear binomials (x_{row(+)} − x_{col(+)}), one for each crossing tile in D.

Theorem (Pipe formula). Q_{r}(x − ^{∘}x) = ∑_{D∈RP(v(r))}(x_{r} − ^{∘}x_{r})^{D∖D_{Hom}}.

This follows from Theorem 5.5. It is geometric in the sense that summands on the righthand side are equivariant classes of coordinate subspaces in a flat (Gröbner) degeneration [KM03a, Theorem B] of the matrix Schubert variety for v(r) (Section 2.1). This is the largest matrix Schubert variety whose quotient modulo the appropriate parabolic subgroup contains the Zelevinsky image of Ω_{r} as an open dense subvariety. Example. Here is a typical pipe dream for the Zelevinsky permutation v(r) from the previous Example, drawn in two ways.

\begin{aligned}& d_{1} & c_{1} & c_{2} & c_{3} & b_{1} & b_{2} & b_{3} & a_{1} \\ a_{1} & + & + & + & + & · & · & + & · \\ b_{1} & + & · & · & · & · & · & · & · \\ b_{2} & + & · & + & · & · & · & · & · \\ b_{3} & + & · & · & · & · & · & · & · \\ c_{1} & · & · & · & · & · & · & · & · \\ c_{2} & · & · & · & · & · & · & · & · \\ c_{3} & · & · & · & · & · & · & · & · \\ d_{1} & · & · & · & · & · & · & · & · \end{aligned} ⟷ [figure: 5; 2; 3; 6; 1; 4; 8; 7] = D

The right-hand diagram depicts the and [figure] tiles, but it omits those parts of pipes below the main antidiagonal, since the “sea” of [figure] tiles there can be confusing to look at.

The elbows have been entirely omitted from the left-hand diagram, but its row and column labels are present. The pipe dream D_{Hom} has tiles precisely in the diagram of v(Hom), where the ∗ entries are in the ratio formula example. The expression (x_{r} − ^{∘}x_{r})^{D∖D_{Hom}} for the above pipe dream D is simply (a_{1} − b_{3})(b_{2} − c_{2}).

For this permutation v(r), moving the ‘+’ in the bc block to either of the two remaining available cells on the antidiagonal of the bc block produces another reduced pipe dream for v(r). Independently, the other ‘+’ can move freely along the antidiagonal on which it sits. None of the ‘+’ entries in D_{Hom} can move. Therefore the right-hand side of the pipe formula becomes a product of two linear forms, namely the sums

((a_{1}−b_{3}) + (b_{1}−b_{2}) + (b_{2}−b_{1}) + (b_{3}−c_{3}) + (c_{1}−c_{2}) + (c_{2}−c_{1}) + (c_{3}−d_{1}))
and ((b_{1}−c_{3}) + (b_{2}−c_{2}) + (b_{3}−c_{1}))

of the binomials associated to cells on the corresponding antidiagonals. Cancellation occurs in the longer of these two linear forms to give (a_{1} − d_{1}), so the product of these two forms returns Q_{r} = (b_{1} + b_{2} + b_{3} − c_{1} − c_{2} − c_{3})(a_{1} − d_{1}) again.

Component formula (lacing diagrams). Double Schubert polynomials are not symmetric in general. Stanley introduced certain symmetrized versions [Sta84] which are now called double Stanley symmetric functions or stable double Schubert polynomials and denoted by F_{w}(X − Y). They are indexed by permutations w and take as arguments a pair of infinite alphabets X and Y. When we evaluate such a symmetric function on a finite alphabet, we mean to set all remaining variables to zero. Stanley functions are produced by an algebraic limiting procedure (Proposition 6.5) from Schubert polynomials; hence the term ‘stable polynomial’. In general, if a double Schubert polynomial 𝔖_{w} and double Stanley function F_{w} are evaluated on the same pair of finite alphabets, the polynomial F_{w}, which is symmetric separately in each of the two alphabets, will tend to have many more terms.

Schubert polynomials and Stanley functions can be defined for any partial permutation w, by which we mean a rectangular matrix filled with zeros except for at most one 1 in each row and column, by canonically extending w to a permutation (Section 2.1). Partial permutations are matrices with entries in the field 𝕜, so it makes sense to say that a list w = (w_{1}, …, w_{n}) of partial permutations lies in Hom, if each w_{i} has size r_{i−1} × r_{i}. Such lists of partial permutation matrices can be identified with nonembedded graphs drawn in the plane, called lacing diagrams, that we define in Section 3. The vertex set of w consists of n + 1 columns of dots, where column i has r_{i} dots. An edge of w connects the dot at height α in column i − 1 to the dot at height β in column i if the αβ entry of w_{i} is 1.

Lacing diagrams come with a natural notion of length derived from the Bruhat order. The minimum possible length for a lacing diagram with rank array r is the codimension of the quiver locus Ω_{r}, which is the total degree of Q_{r}(x − ^{∘}x). Denote by W (r) the set of minimal length lacing diagrams with rank array r (characterized combinatorially in Theorem 3.8).

Since Schubert polynomials can be indexed by partial permutations, we write

𝔖_{w}(x − ^{∘}x) = 𝔖_{w_{1}}(x_{r}^{0}−x_{r}^{1}) ⋯ 𝔖_{w_{n}}(x_{r}^{n−1}−x_{r}^{n})

for products of double Schubert polynomials indexed by the partial permutations in w. Similarly, we have the product notation

F_{w}(x − ^{∘}x) = F_{w_{1}}(x^{0}−x^{1}) ⋯ F_{w_{n}}(x^{n−1}−x^{n})

for double Stanley symmetric functions. Again, 𝔖_{w} and F_{w} take sequences of infinite alphabets as input, but only finitely many variables appear in 𝔖_{w}(x − ^{∘}x), and we are allowed to write F_{w}(x_{r} − ^{∘}x_{r}) if we want to evaluate F_{w} on sequences of finite alphabets.

Theorem (Component formula). Q_{r}(x − ^{∘}x) = ∑_{w∈W (r)} 𝔖_{w}(x − ^{∘}x) = ∑_{w∈W (r)} F_{w}(x_{r} − ^{∘}x_{r}) .

The two versions are Corollary 6.17 and Corollary 6.23. The theorem implies, in particular, that all the extra terms in the Stanley version cancel.

The first sum in this theorem is combinatorially positive in a manner that most directly reflects the geometry of quiver loci. The basic idea is to flatly degenerate the group action of GL under which the quiver locus Ω_{r} is an orbit closure (Section 4). As GL degenerates, so do its orbits, and the flat limits of the orbits are stable under the action of the limiting group. (The general version of this statement is Proposition 4.1; the specific case of interest to us is Proposition 4.5.) The limit need not be irreducible; its components are precisely the closures of orbits (under the limiting group) through minimal length lacing diagrams, which are matrix lists in Hom. This, together with the statement that the components in the degenerate limit of Ω_{r} are generically reduced, is precisely the content of Theorem 6.16. It is our main geometric theorem concerning quiver loci and it immediately implies the Schubert version of the component formula. Example. Continuing with r from (∗), the set W (r) consists of the following three minimal length lacing diagrams, with their partial permutation lists underneath:

([^{1} ^{0} ^{0}] , [\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}], [\begin{matrix}0 \\ 1 \\ 0 \end{matrix}])  ([^{1} ^{0} ^{0}] , [\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}], [\begin{matrix}1 \\ 0 \\ 0 \end{matrix}])  ([^{0} ^{1} ^{0}] , [\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}], [\begin{matrix}1 \\ 0 \\ 0 \end{matrix}])

The two versions (Schubert and Stanley) of the component formula read

Q_{r} = 𝔖_{1243}(b − c) · 𝔖_{213}(c − d) + 𝔖_{2143}(b − c) + 𝔖_{213}(a − b) · 𝔖_{1243}(b − c)
= F_{1243}(b_{3} − c_{3})F_{213}(c_{3} − d_{1}) + F_{2143}(b_{3} − c_{3}) + F_{213}(a_{1} − b_{3})F_{1243}(b_{3} − c_{3})

In the bottom line above, we have written a_{1} = {a_{1}}, b_{3} = {b_{1}, b_{2}, b_{3}}, c_{3} = {c_{1}, c_{2}, c_{3}}, and d_{1} = d_{1} for the finite alphabets corresponding to a, b, c, and d. We have indexed the Schubert polynomials and Stanley functions by permutations instead of partial permutations by completing each partial permutation w to a permutation w̃.

Tableau formula (peelable tableaux and factor sequences). Our previous formulae write quiver polynomials in terms of binomials, double Schubert polynomials (in two completely different ways), and double Stanley symmetric functions. Now we turn to Schur functions. For a list \underline{λ} = (λ_{1}, …, λ_{n}) of partitions, write

s_{\underline{λ}}(x − ^{∘}x) = s_{λ_{1}}(x^{0} − x^{1}) ⋯ s_{λ_{n}}(x^{n−1} − x^{n})

to denote the corresponding product of double Schur functions in infinite alphabets (Section 1.4). Using our Theorem 1.20 to identify our quiver polynomials with those in [BF99], the Main Theorem of Buch and Fulton states that there exist unique integers c_{\underline{λ}}(r) satisfying

Q_{r}(x − ^{∘}x) = ∑_{\underline{λ}} c_{\underline{λ}}(r)s_{\underline{λ}}(x_{r} − ^{∘}x_{r}),

and exhibits an explicit way to generate them. Although this procedure involves negative integers, Buch and Fulton conjectured the positivity of all the quiver constants c_{\underline{λ}}(r). In addition, starting with a rank array r, they produced a concrete recursive method for generating lists of semistandard Young tableaux (Section 7), called factor sequences (Section 8.1). Defining Φ(r) to be the set of factor sequences coming from the rank array r, every list W ∈ Φ(r) of tableaux has an associated list \underline{λ}(W) of partitions. The combinatorial conjecture of [BF99] says that c_{\underline{λ}}(r) counts the number of factor sequences W ∈ Φ(r) of shape \underline{λ}.

The direct connection between factor sequences and the combinatorial geometry of quiver polynomials was from the beginning—and still remains now—a mystery to us. It seems that the question should come down to finding an appropriate geometric explanation for Schur-positivity of Stanley symmetric functions, which is known both combinatorially and algebraically from various points of view. Although we lack a geometric framework, one of these other points of view, namely that of Demazure characters (Section 7.1), still allows us to deduce a combinatorial formula for the quiver constants c_{\underline{λ}}(r).

The argument is based on the fact that every Stanley function F_{w} expands as a sum ∑_{λ} α_{w}^{λ}s_{λ} of Schur functions s_{λ} with nonnegative integer Stanley coefficients α_{w}^{λ} (Section 6.1). The main point, stated precisely in Theorem 7.14, is that every quiver constant is a Stanley coefficient for the Zelevinsky permutation: c_{\underline{λ}}(r) = α_{v(r)}^{λ} for a certain partition λ.

It is known that every Stanley coefficient counts a set of peelable tableaux (Section 7.4). Our case focuses on the set Peel(D_{r}) of peelable tableaux for the diagram D_{r} of the Zelevinsky permutation v(r). A semistandard tableau P lies in Peel(D_{r}) if it satisfies a readily checked, mildly recursive condition. The shape of every such tableau contains the partition whose shape is the diagram D_{Hom} of v(Hom), so removing D_{Hom} leaves a skew tableau P − D_{Hom}. The connected components of this skew tableau form a list Ψ_{r}(P) of tableaux, read northeast to southwest (Definition 8.13), whose list of shapes we denote by \underline{λ}(P). Theorem (Tableau formula; Buch–Fulton factor sequence conjecture).

Q_{r}(x − ^{∘}x) = ∑_{P ∈Peel(r)} s_{\underline{λ}(P)}(x_{r} − ^{∘}x_{r})

= ∑_{W ∈Φ(r)} s_{\underline{λ}(W)}(x_{r} − ^{∘}x_{r}).

The two formulae in this theorem come from Theorem 7.21 and Corollary 8.23 (see Theorem 7.10, as well, which says that our c_{\underline{λ}}(r) agree with those in [BF99]). In contrast to the Schubert and Stanley versions of the component formula, the peelable and factor sequence versions of the tableau formula are equal summand by summand; this is immediate from Theorem 8.22, which says simply that P ↦ Ψ_{r}(P) is a bijection from peelable tableaux to factor sequences. We shall mention more about the proof of the tableau formula later in the Introduction. Example. The set Peel(D_{r}) of peelable tableaux for D_{r} consists of four tableaux:

1 1 1 1 7 2 4 3 4  1 1 1 1 2 4 7 3 4  1 1 1 1 2 4 3 7 4  1 1 1 1 2 4 3 4 7

Hence, under the bijection Ψ_{r} that acts by removing entries whose locations lie in the diagram of v(Hom) (the ∗ entries in the ratio formula example), there are four factor sequences:

( 7 , 4 , ∅)  (∅ , 4 7 , ∅)  (∅ , 4 7 , ∅)  (∅ , 4 , 7 )

Using finite alphabets as in the component formula example, the tableau formula for Q_{r} is

Q_{r} = s (a_{1} − b_{3})s (b_{3} − c_{3}) + s (b_{3} − c_{3}) + s (c_{3} − d_{1}) + s (b_{3} − c_{3})s (c_{3} − d_{1}).

Proofs via double versions and limits. The formulae we presented above are all for ordinary quiver polynomials. One of the innovations in this paper is the idea of working with certain double versions of quiver polynomials, defined via the ratio formula. We clarify their combinatorics by applying limits that combine the way Stanley symmetric functions come from Schubert polynomials with the way Buch’s series P_{r} come from quiver K-classes [Buc02, Section 4]. Our need for these limits arises because we require a component formula with symmetric summands. Our need for double versions arises because at a key stage we must set the ^{∘}x variables to zero without affecting the x variables.

The doubling construction simply replaces the sequence ^{∘}x of alphabets with a new sequence ^{∘}y consisting of variables independent of the x variables. All of the definitions go through verbatim after replacing each symbol ^{∘}x or x^{j} that has a minus sign in front of it by the corresponding y symbol. For example, we define the double quiver polynomial Q_{r}(x−^{∘}y) by making this replacement in the ratio formula (Definition 2.5). Given this definition, the content of the ratio formula for ordinary quiver polynomials is that specializing ^{∘}y = ^{∘}x in the ratio Q_{r}(x − ^{∘}y) of double Schubert polynomials yields the multidegree of the quiver locus Ω_{r} inside Hom. Interpreting the numerator and denominator of Q_{r}(x − ^{∘}y) as multidegrees of matrix Schubert varieties (Section 2.1), this is accomplished in Section 2.3 using the Zelevinsky map from Section 1.3, which takes the quiver locus Ω_{r} isomorphically to the intersection of the matrix Schubert variety \overline{X}_{v(r)} with an appropriate “opposite big cell”.

The Zelevinsky map was introduced in a slightly different form in [Zel85]. The fact that it is a scheme-theoretic isomorphism is equivalent to the main result in [LM98]. However, since we need additionally the explicit combinatorial connection to Zelevinsky permutations, we derive a new proof (Theorem 1.14), which happens to be quite elementary. It does not even rely on the primeness of the ideal I_{r} defining Ω_{r} as a subscheme of Hom.

The double version of the pipe formula, Theorem 5.5, is automatic from the formula for double Schubert polynomials in [FK96], as we explain in Section 5.1.

Unlike the ratio and pipe formulae, simply replacing ^{∘}x by ^{∘}y in the component formula almost always yields a provably false statement (Remark 6.26). On the other hand, it turns out not to be useful to have a double component formula in which the summands are not symmetric, since (as we shall explain shortly) the point is to decompose the summands in terms of Schur polynomials. To rectify both the symmetry problem and the obstruction to doubling the variables, we resort to algebraic limits (reviewed in Section 6.1) of the sort used to construct Stanley functions from Schubert polynomials. The essential point is that when a rank array r is replaced by rank array m + r, in which each r_{ij} is increased by m, nothing fundamental changes about the geometry, algebra, or combinatorics of the quiver locus Ω_{r}. The development of this idea culminates in the following master component formula (Theorem 6.20), which is arguably the single most important result in the paper. Theorem (Stable double component formula).

F_{r}(x − ^{∘}y) := lim_{m→∞} Q_{m+r}(x − ^{∘}y) = ∑_{w∈W (r)} F_{w}(x − ^{∘}y).

The most important stability result on the way to this formula is Proposition 4.13: the lacing diagrams indexing the components of the quiver degenerations for the rank arrays r and m + r are in canonical bijection, and furthermore the multiplicities of the corresponding components are equal. Taking multidegrees, this immediately implies the positivity part of the Buch–Fulton conjecture for quiver constants geometrically, using none of the four formulae (Remark 4.15). Further expository details concerning the geometric, algebraic, and combinatorial roles of limits and stability in this paper, especially the manner in which they enter into the proof of the above formula, are included in the introduction to Section 6.

Why do we need double and stable versions? At a crucial point in our derivation of the peelable tableau formula from a component formula (see Remark 7.15), we need certain key polynomials to equal the products of Schur polynomials that they a priori only approximate. We check this by applying the theory of Demazure characters via isobaric divided differences (Section 7.2). But in the literature, these have only been developed for ordinary (not double) Schubert polynomials, so we must work with ratios 𝔖_{v(r)}(x)/𝔖_{v(Hom)}(x) = Q_{r}(x) of ordinary Schubert polynomials obtained from double quiver polynomials by setting ^{∘}y = 0.

To be more precise, consider the two sides of the equation

\frac{𝔖_{v(r)}(x)}{𝔖_{v(Hom)}(x)} = ∑_{w∈W (r)} F_{w}(x_{r}),

which we prove in Proposition 7.13 from the stable double component formula. Both sides expand into sums of products s_{\underline{λ}}(x_{r}) of ordinary Schur polynomials, the left side by Demazure characters, and the right side by using the fact that each summand is a product of symmetric polynomials in the sequence x_{r}. This formula simultaneously explains our need for doubled alphabets as well as for stabilization: Schur-expansion of the left-hand side demands double versions because it requires ^{∘}y = 0, while Schur-expansion of the right-hand side demands stability because it requires the summands to be symmetric functions.

We show in Theorem 7.10 that the coefficients in the Schur-expansion of the right-hand side are the quiver constants from [BF99], while the above Demazure character argument implies that the coefficients on the left-hand side are Stanley coefficients (Theorem 7.14). These count peelable tableaux, proving the peelable tableau formula. The factor sequence conjecture follows from the peelable tableau formula via the bijection Ψ_{r}. The proof of this bijection in Section 8 is a complex but essentially elementary computation. It works by breaking the diagram D_{r} into pieces small enough so that putting them back together in different ways yields peelable tableaux on the one hand, and factor sequences on the other.

Related notions and extensions. Quiver loci have been studied by a number of authors in their roles as universal degeneracy loci for vector bundle morphisms, as generalizations of Schubert varieties, and in representation theory of quivers; see [AD80, ADK81, LM98, FP98, Ful99, BF99, FR02b], for a sample. The particular rank conditions we consider here appeared first in [BF99] as the culmination of an increasingly general progression of degeneracy loci. Snapshots from this progression include the following; for more on the history, see [FP98].

• The case of two vector bundles and a fixed map required to have rank bounded by a given integer is known as the Giambelli–Thom–Porteous formula; see [Ful98, Chapter 14.4]. In this case both the ratio formula and the component formula reduce directly to the Giambelli–Thom–Porteous formula: the denominator in the ratio formula equals 1, while its numerator is the appropriate Schur polynomial; and there is just one lacing diagram with no crossing laces.

• Given two filtered vector bundles, there are degeneracy loci defined by Schubert conditions bounding the ranks of induced morphisms from subbundles in the filtration of the source to quotients by those in the target. The resulting universal formulae are double Schubert polynomials [Ful92].

• The same Schubert conditions can be placed on a quiver like those considered here, with 2n bundles of ranks increasing from 1 up to n and then decreasing from n down to 1. The degeneracy locus formula produces polynomials that Fulton called ‘universal Schubert polynomials’ [Ful99], because they specialize to quantum and double Schubert polynomials. We propose to call them Fulton polynomials. Results similar to the ones we prove here have been obtained independently for the special case of Fulton polynomials in [BKTY02].

One can consider even more general quiver loci and degeneracy loci, for arbitrary quivers and arbitrary rank conditions. We expect that the resulting quiver polynomials should have interesting combinatorial descriptions, at least for finite type quivers. Some indications of this come from [FR02b], where the same general idea of reducing to the equivariant study of quiver loci also appears (but the specifics differ in what kinds of statements are made concerning quiver polynomials), along with Rimányi’s Thom polynomial proof of the component formula [BFR03], which he produced in response to seeing the formula.

For nilpotent cyclic quivers the orbit degeneration works just as for the type A_{n+1} quiver considered in this paper, and the Zelevinsky map can be replaced by Lusztig’s embedding of a nilpotent cyclic quiver into a partial flag variety [Lus90]. The technique of orbit degeneration might also extend to arbitrary finite type quivers, but Zelevinsky maps do not extend in any straightforward way. One of the advantages of orbit degeneration is that it not only implies existence of a combinatorial formula, but provides strong hints (Proposition 4.11) as to the format of such a formula. Thus, in contrast to [BF99, p. 668], where much of the work lay in “discovering the shape of the formula”, the geometry of degeneration provides a blueprint automatically.

Among the cohomological statements in this paper lurks a single result in K-theory: Theorem 2.7 gives a combinatorial formula for quiver K-classes as ratios of double Grothendieck polynomials. This K-theoretic analogue of the ratio formula (which is cohomological) immediately implies the K-theoretic analogue of our pipe formula by [FK94] (see [KM03b, Section 5]). There is also a K-theoretic generalization of our component formula, proved independently in two papers subsequent to this one [Buc03, Mil03]. It implies Buch’s conjecture [Buc02] that the K-theoretic analogues of the quiver constants c_{\underline{λ}}(r) exhibit a certain sign alternation, generalizing the positivity of c_{\underline{λ}}(r).

It is an interesting problem to find a K-theoretic analogue of the factor sequence tableau formula. Substantial work in this direction has already been done by Buch [Buc02]. In analogy with our proof of the cohomological Buch–Fulton factor sequence conjecture, a complete combinatorial description of its K-theoretic analogue will involve a better understanding of how stable Grothendieck polynomials indexed by permutations expand into stable Grothendieck polynomials indexed by partitions. Lascoux has given an algorithm for this expansion based on a transition formula for Grothendieck polynomials [Las01].

A note on the field 𝕜. Although to simplify the exposition we use language at times as if the field 𝕜 were algebraically closed, all of our results hold for an arbitrary field 𝕜. In fact the schemes we consider are all defined over the integers ℤ, and our geometric statements concerning flat (Gröbner) degenerations work in that context.

Section 1. Geometry of quiver loci

1.1. Quiver loci and ideals

Fix a sequence V = (V_{0}, V_{1}, …, V_{n}) of vector spaces with dim(V_{i}) = r_{i}, and denote by Hom the variety Hom(V_{0}, V_{1}) × ⋯ × Hom(V_{n−1}, V_{n}) of type A_{n+1} quiver representations on V. That is, Hom equals the vector space of sequences

ϕ : V_{0} ⟶^{ϕ_{1}} V_{1} ⟶^{ϕ_{2}} ⋯ ⟶^{ϕ_{n−1}} V_{n−1} ⟶^{ϕ_{n}} V_{n}

of linear transformations. By convention, set V_{−1} = 0 = V_{n+1}, and ϕ_{0} = 0 = ϕ_{n+1}. Once and for all fix a basis^{1} for each vector space V_{i}, and express elements of V_{i} as row vectors of length r_{i}. Doing so identifies each map ϕ_{i} in a quiver representation ϕ with a matrix of size r_{i−1} × r_{i}. Thus the coordinate ring 𝕜[Hom] becomes a polynomial ring in variables (f^{1}_{αβ}), …, (f^{n}_{αβ}), where the i^{th} index β and the (i + 1)^{st} index α run from 1 to r_{i}. Let Φ be the generic quiver representation, in which the entries in the matrices Φ_{i} : V_{i−1} → V_{i} are the variables f^{i}_{αβ}.

Definition 1.1. Given an array r = (r_{ij})_{0≤i≤j≤n} of nonnegative integers, the quiver locus Ω_{r} is the zero scheme of the quiver ideal I_{r} ⊂ 𝕜[Hom] generated by the union over all i < j of the minors of size (1 + r_{ij}) in the product Φ_{i+1} ⋯ Φ_{j} of matrices:

I_{r} = ⟨minors of size (1 + r_{ij}) in Φ_{i+1} ⋯ Φ_{j} for i < j⟩.

Thus the quiver locus Ω_{r} gives a natural scheme structure to the set of quiver representations whose composite maps V_{i} → V_{j} have rank at most r_{ij} for all i < j.

Two quiver representations (V, ϕ) and (W, ψ) on sequences V and W of n vector spaces are isomorphic if there are linear isomorphisms η_{i} : V_{i} → W_{i} for i = 0, …, n commuting with ϕ and ψ. Also, we can take the direct sum V ⊕ W in the obvious manner. Quiver representations that cannot be expressed nontrivially as direct sums are called indecomposable. Whenever 0 ≤ p ≤ q ≤ n, there is an (obviously) indecomposable representation

I_{p,q} : 0 → ⋯ → 0 → 𝕜_{p} = ⋯ = 𝕜_{q} → 0 → ⋯ → 0

having copies of the field 𝕜 in spots between p and q, with identity maps between them and zeros elsewhere. Here is a standard result, c.f. [LM98, Section 1.1], generalizing the rank–nullity theorem from linear algebra.

Proposition 1.2. Every indecomposable quiver representation is isomorphic to some I_{p,q}, and every quiver representation ϕ ∈ Hom is isomorphic to a direct sum of these.

The space Hom of quiver representations carries an action of the group

(1.1)  GL^{2} = GL(V_{0})^{2} × GL(V_{1})^{2} × ⋯ × GL(V_{n−1})^{2} × GL(V_{n})^{2}

of linear transformations. Specifically, if we think of elements in each V_{i} as row vectors and writing GL(V_{i})^{2} = \overleftarrow{GL}(V_{i}) × \overrightarrow{GL}(V_{i}) for i = 0, …, n, the factor \overleftarrow{GL}(V_{i}) acts by inverse multiplication on the right of Hom(V_{i−1}, V_{i}), while the factor \overrightarrow{GL}(V_{i}) acts by multiplication on the left of Hom(V_{i}, V_{i+1}). The group

GL = GL(V_{0}) × ⋯ × GL(V_{n})

embeds diagonally inside GL^{2}, so that γ = (γ_{0}, …, γ_{n}) ∈ GL acts by

γ · ϕ = ( …, γ_{i−1}ϕ_{i}γ_{i}^{−1}, γ_{i}ϕ_{i+1}γ_{i+1}^{−1}, …).

1This is not so bad, since we shall in due course require a maximal torus in GL(V_{0}) × ⋯ × GL(V_{n}).

This action of GL preserves ranks, in the sense that GL · Ω_{r} = Ω_{r} for any ranks r, because γ_{i}^{−1} cancels with γ_{i} when the maps in γ · ϕ are composed.

Lemma 1.3. The group GL has finitely many orbits on Hom, and every quiver locus Ω_{r} is supported on a union of closures of such orbits.

Proof. Since GL · Ω_{r} = Ω_{r}, quiver loci are unions of orbit closures for GL, so it is enough to prove there are finitely many orbits. Proposition 1.2 implies that given any quiver representation ϕ ∈ Hom, we can choose new bases for V_{0}, …, V_{n} in which ϕ is expressed as a list of partial permutation matrices (all zero entries, except for at most one 1 in each row and column). Hence ϕ lies in the GL orbit of one of the finitely many lists of partial permutation matrices in Hom. □

Definition 1.1 makes no assumptions about the array r of nonnegative integers, but only certain arrays can actually occur as ranks of quivers. More precisely, associated to each quiver representation ϕ ∈ Hom is its rank array r(ϕ), whose nonnegative integer entries are given by the ranks of composite maps V_{i} → V_{j}:

r_{ij}(ϕ) = rank(ϕ_{i+1}ϕ_{i+2} ⋯ ϕ_{j}) for i < j,

and r_{ii}(ϕ) = r_{i} for i = 0, …, n. It is a consequence of the discussion above that the support of a quiver locus Ω_{r} is irreducible if and only if r = r(ϕ) occurs as the rank array of some quiver representation ϕ ∈ Hom, and this happens if and only if Ω_{r} equals the closure of the orbit of GL through ϕ.

Convention 1.4. We make the convention here, once and for all, that in this paper we consider only rank arrays r that can occur, so r = r(ϕ) for some ϕ ∈ Ω_{r}.

That being said, we remark that this assumption is unnecessary in the part of Section 1.4 preceding Theorem 1.20, as well as in Section 4.

1.2. Rank, rectangle, and lace arrays

Given a rank array r = r(ϕ) that can occur, Proposition 1.2 implies that there exists a unique array of nonnegative integers s = (s_{ij})_{0≤i≤j≤n} such that ϕ = ⨁_{p≤q} I_{p,q}^{⊕s_{pq}}. In other words, the ranks r can occur if and only if there is an associated array s. We call this data the lace array; the nomenclature is explained by Lemma 3.2, and the symbol ‘s’ recalls that our construction is based on the ‘strands’ appearing in [AD80]. Taking ranks, we find that

(1.2)  r_{ij} = ∑_{{k≤i \atop j≤m}} s_{km} for i ≤ j.

The (transposed^{2}) Buch–Fulton rectangle array is the array of rectangles (R_{ij}) for 0 ≤ i < j ≤ n, such that R_{ij} is the rectangle of height r_{i,j−1} − r_{ij} and width r_{i+1,j} − r_{ij}.

2The rectangle R_{ij} of [BF99] is the transpose of ours.

Example 1.5. Consider the rank array r = (r_{ij}), its lace array s = (s_{ij}), and its rectangle array R = (R_{ij}), which we depict as follows.

r = \begin{aligned}3 & 2 & 1 & 0 & i⧸j \\ & & & 2 & 0 \\ & & 3 & 2 & 1 \\ & 4 & 2 & 1 & 2 \\ 3 & 2 & 1 & 0 & 3 \end{aligned}  s = \begin{aligned}3 & 2 & 1 & 0 & i⧸j \\ & & & 0 & 0 \\ & & 0 & 1 & 1 \\ & 1 & 0 & 1 & 2 \\ 1 & 1 & 1 & 0 & 3 \end{aligned}  R = \begin{aligned}2 & 1 & 0 & i⧸j \\ & & & 1 \\ & & & 2 \\ & & & 3 \end{aligned}

The relation (1.2) says that an entry of r is the sum of the entries in s that are weakly southeast of the corresponding location. The height of R_{ij} is obtained by subtracting the entry r_{ij} from the one above it, while the width of R_{ij} is obtained by subtracting the entry r_{ij} from the one to its left. □

It follows from the definition of R_{ij} that

(1.3)  ∑_{k≥j} height(R_{ik}) = r_{i,j−1} − r_{i,n} ≤ r_{i,j−1}  for all i
(1.4)  ∑_{ℓ≤i} width(R_{ℓj}) = r_{i+1,j} − r_{0,j} ≤ r_{i+1,j}  for all j.

(This will be applied in Proposition 8.12.) The relation (1.2) can be inverted to obtain

(1.5)  s_{ij} = r_{ij} − r_{i−1,j} − r_{i,j+1} + r_{i−1,j+1}

for i ≤ j, where r_{ij} = 0 if i and j do not both lie between 0 and n. Therefore the rank array r can occur for V if and only if r_{ii} = r_{i} for i = 0, …, n and

(1.6)  r_{ij} − r_{i−1,j} − r_{i,j+1} + r_{i−1,j+1} ≥ 0

for i ≤ j, the left-hand side being simply s_{ij}. We shall interchangeably use a lace array s or its corresponding rank array r to describe a given irreducible quiver locus.

It follows from (1.6) that

(1.7) height(R_{ij}) decreases as i decreases. (1.8) width(R_{ij}) decreases as j increases.

Given a rank array r or equivalently a lace array s, we shall construct a permutation v(r) ∈ S_{d}, where d = r_{0} + ⋯ + r_{n}. In general, any matrix in the space M_{d} of d × d matrices comes with a decomposition into block rows of heights r_{0}, …, r_{n} (from top to bottom) and block columns of lengths r_{n}, …, r_{0} (from left to right). Note that our indexing convention may be unexpected, with the square blocks lying along the main block antidiagonal rather than on the diagonal as usual. With these conventions, the i^{th} block column refers to the block column of length r_{i}, which sits i blocks from the right.

We draw the matrix for the permutation w with a symbol × (instead of a 1) at each position (q, w(q)) and zeros elsewhere.

Proposition 1.6. Given a rank array r for V, there exists a unique element v(Ω_{r}) = v(r) ∈ S_{d} satisfying the following conditions. Consider the block in the i^{th} block column and j^{th} block row.

1. If i ≤ j (that is, the block sits on or below the main block antidiagonal) then the number of × entries in that block equals s_{ij}.

2. If i = j+1 (that is, the block sits on the main block superantidiagonal) then the number of × entries in that block equals r_{j,j+1}.

\begin{matrix}8 & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & × & · & · & · & · \\ 9 & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & · & × & · & · & · \\ 4 & ∗ & ∗ & ∗ & × & · & · & · & · & · & · & · & · \\ 5 & ∗ & ∗ & ∗ & · & × & · & · & · & · & · & · & · \\ 11 & ∗ & ∗ & ∗ & · & · & □ & □ & · & · & □ & × & · \\ 1 & × & · & · & · & · & · & · & · & · & · & · & · \\ 2 & · & × & · & · & · & · & · & · & · & · & · & · \\ 6 & · & · & □ & · & · & × & · & · & · & · & · & · \\ 12 & · & · & □ & · & · & · & □ & · & · & □ & · & × \\ 3 & · & · & × & · & · & · & · & · & · & · & · & · \\ 7 & · & · & · & · & · & · & × & · & · & · & · & · \\ 10 & · & · & · & · & · & · & · & · & · & × & · & · \end{matrix}

Figure 1. Zelevinsky permutation and its diagram

3. If i ≥ j + 2 (that is, the block lies strictly above the main block superantidiagonal) then there are no × entries in that block.

4. Within every block row or block column, the × entries proceed from northwest to southeast, that is, no × entry is northeast of another × entry.

Definition 1.7. v(r) is the Zelevinsky permutation for the rank array r.

Proof. We must show that the number of × entries in any block row, as dictated by conditions 1–3, equals the height of that block row (and transposed for columns), since condition 4 then stipulates uniquely how to arrange the × entries within each block. In other words the height r_{j} of the j^{th} block row must equal the number r_{j,j+1} of × entries in the superantidiagonal block in that block row, plus the sum ∑_{i} s_{ij} of the number of × entries in the rest of the blocks in that block row (and a similar statement must hold for block columns). These statements follow from (1.2). □

The diagram D(v) of a permutation v ∈ S_{d}, which by definition consists of those cells in the d × d grid having no × in v due north or due west of it, refines the data contained in the Buch–Fulton rectangle array [BF99]. The following Lemma is a straightforward consequence of the definition of the Zelevinsky permutation v(r) and equation (1.5); to simplify notation, we often write D_{r} = D(v(r)).

Lemma 1.8. In any block of the diagram D_{r} = D(v(r)), the cells form a rectangle justified in the southeast corner of the block. If the block is above the superantidiagonal, this rectangle consists of all the cells in the block. If the block is on or below the superantidiagonal, say in the i^{th} block column and j^{th} block row, this rectangle is the Buch–Fulton rectangle R_{i−1,j+1}.

Example 1.9. Let r, s, R be as in Example 1.5. The Zelevinsky permutation is

v(r) = (\begin{matrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ 8 & 9 & 4 & 5 & 11 & 1 & 2 & 6 & 12 & 3 & 7 & 10 \end{matrix}) ,

whose permutation matrix is indicated by × entries in Fig. 1, and whose diagram D_{r} = D(v(r)) is indicated by the union of the ∗ and □ entries there. □

Definition 1.10. Consider a fixed dimension vector (r_{0}, r_{1}, …, r_{n}) and a varying rank array r with that dimension vector.

1. The diagram D_{Hom} := D(v(Hom)) for the quiver locus Hom consists of all cells above the block superantidiagonal. Thus D_{r} ⊃ D_{Hom} for all occurring rank arrays r. Define the (skew) diagram D_{r}^{□} for r as the difference

D_{r}^{□} = D_{r} ∖ D_{Hom}.

In our depictions of D_{r}, each cell of D_{r}^{□} is indicated by a box □, while each cell of D_{Hom} is indicated by an asterisk ∗ (see Fig. 1, for example).

2. For the zero quiver Ω_{0}, whose rank array r_{0} is zero except for the r_{ii} entries, the diagram D(Ω_{0}) consists of all locations strictly above the antidiagonal.

1.3. The Zelevinsky map

Let GL_{d} be the invertible d × d matrices. Denote by P ⊂ GL_{d} the parabolic subgroup of block lower triangular matrices, where the diagonal blocks have sizes r_{0}, …, r_{n}, and let B_{+} be the group of upper triangular matrices.

The group P acts by multiplication on the left of GL_{d}, and the quotient P \GL_{d} is the manifold of partial flags with dimension jumps r_{0}, …, r_{n}. Schubert varieties X_{v} in P \GL_{d} are orbit closures for the (left) action of B_{+} by inverse right multiplication. In writing X_{v} ⊆ P \GL_{d} we shall always assume that the permutation v ∈ S_{d} has minimal length in its right coset (S_{r_{0}} × ⋯ × S_{r_{n}})v, which means that v has no descents within a block. Graphically, this condition is reflected in the permutation matrix for v by saying that no × entry is northeast of another in the same block row. Observe that this condition holds by definition for Zelevinsky permutations v(r).

The preimage in GL_{d} of a Schubert variety X_{v} ⊆ P \GL_{d} is the closure in GL_{d} of the double coset P vB_{+}. The closure \overline{P vB_{+}} of that inside M_{d} is the ‘matrix Schubert variety’ \overline{X}_{v}, whose definition we now recall. Since we shall need matrix Schubert varieties in the slightly more general setting of partial permutations later on, we use language here compatible with that level of generality.

Definition 1.11 ([Ful92]). Let w be a partial permutation of size k × ℓ, meaning that w is a matrix with k rows and ℓ columns having all entries equal to 0 except for at most one entry equal to 1 in each row and column. Let M_{kℓ} be the vector space of k × ℓ matrices. The matrix Schubert variety \overline{X}_{w} is the subvariety

\overline{X}_{w} = \{Z ∈ M_{kℓ} | rank(Z_{q×p}) ≤ rank(w_{q×p}) for all q and p\}

inside M_{kℓ}, where Z_{q×p} consists of the top q rows and left p columns of Z.

If v ∈ S_{d} is a permutation, we do not require v to be a minimal element in its coset of S_{r_{0}} × ⋯ × S_{r_{n}} when we write \overline{X}_{v} (as opposed to X_{v}); however, the matrix Schubert variety will fail to be a P × B_{+} orbit closure in M_{d} unless v is minimal.

Let Y_{0} be the variety of all matrices in GL_{d} whose antidiagonal blocks are all identity matrices, and whose other blocks below the antidiagonal are zero. Thus Y_{0} is obtained from the unipotent radical U(P_{+}) of the block upper-triangular subgroup P_{+} by a global left-to-right reflection followed by block left-to-right reflection. This has the net effect of reversing the order of the block columns of U(P_{+}). If w_{0} is the long element in S_{d}, represented as the antidiagonal permutation matrix, and w_{0} is the block long element, with antidiagonal permutation matrices in each diagonal block, then Y_{0} = U(P_{+})w_{0}w_{0}. The variety Y_{0} maps isomorphically (scheme-theoretically) to the opposite big cell U_{0} in P \GL_{d} under projection modulo P. In other words, U_{0} → Y_{0} is a section of the projection GL_{d} → P \GL_{d} over the open set U_{0}.

Using the isomorphism U_{0} ≅ Y_{0}, the intersection X_{v} ∩ U_{0} of a Schubert variety in P \GL_{d} with the opposite big cell is a closed subvariety Y_{v} of Y_{0} (we assume v is minimal in its coset when we write Y_{v}). It will be important for Theorem 1.14 to make the (standard) comparison between the equations defining Y_{v} inside Y_{0} and the equations defining the corresponding matrix Schubert variety \overline{X}_{v} inside M_{d}.

Lemma 1.12. The ideal of Y_{v} as a subscheme of Y_{0} is obtained from the ideal of \overline{X}_{v} as a subscheme of M_{d} by setting all variables on the diagonal of each antidiagonal block equal to 1 and all other variables in or below the block antidiagonal to zero.

Proof. The statement is equivalent to its geometric version, which says that Y_{v} equals the scheme-theoretic intersection of \overline{X}_{v} with Y_{0}. To prove this version, note that \overline{X}_{v} ∩ Y_{0} = (\overline{X}_{v}∩GL_{d})∩Y_{0}, and that \overline{X}_{v}∩GL_{d} projects to X_{v} as a fiber bundle over X_{v} with fiber P. The intersection of \overline{X}_{v} ∩ GL_{d} with the image Y_{0} of the section U_{0} → GL_{d} is scheme-theoretically the image Y_{v} of the section restricted to X_{v} ∩ U_{0}. □

Definition 1.13. The Zelevinsky map Z takes ϕ ∈ Hom to the block matrix

(1.9)  (ϕ_{1}, ϕ_{2}, …, ϕ_{n}) {Z \atop ⟼} [ \begin{matrix}0 & & 0 & ϕ_{1} & 1 \\ 0 & & ϕ_{2} & 1 & 0 \\ 0 & ... & 1 & 0 & 0 \\ ϕ_{n} & ... & 0 & 0 & 0 \\ 1 & & 0 & 0 & 0 \end{matrix} ] .

Zelevinsky’s original map [Zel85] sent each ϕ ∈ Hom to the matrix (Z(ϕ)w_{0}w_{0})^{−1}, which is obtained from Z(−ϕ) by reversing the block columns and taking the inverse. Zelevinsky proved that it was bijective. The following theorem is therefore equivalent to the main theorem in [LM98], which proved using the primality of ideals of Plücker relations that Zelevinsky’s original map is an isomorphism of schemes. The simpler form Z of Zelevinsky’s map allows us to prove that the relevant ideals are equal without assuming that either is prime; more importantly, it connects quiver loci to the explicit combinatorics of matrix Schubert varieties for Zelevinsky permutations, as we require later.

Theorem 1.14. The Zelevinsky map Z induces a scheme isomorphism from each quiver locus Ω_{r} to the closed subvariety Y_{v(r)} inside the Schubert subvariety X_{v(r)} of the partial flag manifold P \GL_{d}. In other words, Z(Ω_{r}) = Y_{v(r)} as subschemes of Y_{0}.

Proof. Both schemes Z(Ω_{r}) and Y_{v(r)} are contained in Z(Hom): for Z(Ω_{r}) this is by definition, and for Y_{v(r)} this is because the diagram of v(r) contains the union of all blocks strictly above the block superantidiagonal, whence every coordinate above the block superantidiagonal is zero on Y_{v(r)}. Since the Zelevinsky map Ω_{r} → Z(Ω_{r}) is obviously an isomorphism of schemes, we must show that Z(Ω_{r}) is defined by the same equations in 𝕜[Z(Hom)] defining Y_{v(r)}.

The set-theoretic description of \overline{X}_{v} in Definition 1.11 implies that the Schubert determinantal ideal I(\overline{X}_{v}) ⊆ 𝕜[M_{d}] contains the union (over q, p = 1, …, d) of minors with size 1 + rank(v_{q×p}) in the northwest q × p submatrix of the d × d matrix Φ of variables. In fact, using Fulton’s ‘essential set’ [Ful92, Section 3], the ideal I(\overline{X}_{v}) is generated by those minors arising from cells (q, p) at the southeast corner of some block, along with all the variables strictly above the block superantidiagonal.

Consider a box (q, p) at the southeast corner of B_{i+1,j−1}, the intersection of block column i + 1 and block row j − 1, so that by definition of Zelevinsky permutation,

(1.10)  rank(v(r)_{q×p}) = ∑_{{α>i \atop β<j}} s_{αβ} + ∑_{k=i+1}^{j} r_{k−1,k}.

Lemma 1.15. The number rank(v(r)_{q×p}) in (1.10) equals r_{ij} + ∑_{k=i+1}^{j−1} r_{k}.

Proof. The coefficient on s_{αβ} in r_{ij} + ∑_{k=i+1}^{j−1} r_{k} is the number of elements in {r_{ij}} ∪ {r_{i+1,i+1}, …, r_{j−1,j−1}} that are weakly northwest of r_{αβ} in the rank array r (when the array r is oriented so that its southeast corner is r_{0n}). This number equals the number of elements in {r_{i,i+1}, …, r_{j−1,j}} that are weakly northwest of r_{αβ}, unless r_{αβ} happens to lie strictly north and strictly west of r_{ij}, in which case we get one fewer. This one fewer is exactly made up by the sum of entries from s in (1.10). □

Resuming the proof of Theorem 1.14, we set the appropriate variables in the generic matrix Φ to 0 or 1 by Lemma 1.12, and consider the equations coming from the northwest q × p submatrix for (q, p) in the southeast corner of B_{i+1,j−1}. Since Y_{v(r)} ⊆ Z(Hom), these equations are minors of (1.9). In particular, using Lemma 1.15, and assuming that i, j ∈ {0, …, n}, we find that these equations in 𝕜[Z(Hom)] are the minors of size 1+u+r_{ij} in the generic (u + r_{i}) × (u + r_{i}) block matrix

[ \begin{matrix}0 & 0 & & 0 & ϕ_{i+1} \\ 0 & 0 & & ϕ_{i+2} & 1 \\ 0 & 0 & ... & 1 & 0 \\ 0 & ϕ_{j−1} & ... & 0 & 0 \\ ϕ_{j} & 1 & & 0 & 0 \end{matrix} ] , (1.11)

where u = ∑_{k=i+1}^{j−1} r_{k} is the sum of the ranks of the subantidiagonal 1 blocks. The ideal generated by these minors of size 1 + u + r_{ij} is preserved under multiplication of (1.11) by any matrix in SL_{u+r_{i}}(𝕜[Z(Hom)]). In particular, multiply (1.11) on the left by

[ \begin{matrix}1 & −ϕ_{i+1} & ϕ_{i+1}ϕ_{i+2} & ⋯ & ±ϕ_{i+1,j−2} & ∓ϕ_{i+1,j−1} \\ & 1 \\ & & 1 \\ & & & … \\ & & & & 1 \\ & & & & & 1 \end{matrix} ] ,

where ϕ_{i+1,k} = ϕ_{i+1} ⋯ ϕ_{k} for i + 1 ≤ k. The result agrees with (1.11) except in its first block row, which has left block (−1)^{j−1−i}ϕ_{i+1} ⋯ ϕ_{j} and all other blocks zero. Therefore the minors coming from the southeast corner (q, p) of the block B_{i+1,j−1} generate the same ideal in 𝕜[Z(Hom)] as the size 1 + r_{ij} minors of ϕ_{i+1} ⋯ ϕ_{j}, for all i ≤ j. We conclude that the ideals of Z(Ω_{r}) and Y_{v(r)} inside 𝕜[Z(Hom)] coincide. □

The proof of Theorem 1.14 never uses the fact that the minors vanishing on matrix Schubert varieties generate a prime ideal. Nonetheless, they do [Ful92, KM03a], and in fact one can conclude much more by citing (as in [LM98]) other properties of Schubert varieties from [Ram85, RR85], and using Lemma 1.12.

Corollary 1.16 ([LM98, AD80]). Irreducible quiver loci Ω_{r} are reduced, normal, and Cohen–Macaulay with rational singularities. The total area

d(r) = ℓ(v(r)) − ℓ(v(Hom)) = ∑_{0≤k≤i<j≤m≤n} s_{k,j−1}s_{i+1,m}

of the rectangles R_{ij} equals the codimension of Ω_{r} in Hom.

Remark 1.17. One can prove even more simply that the Zelevinsky map Z induces an isomorphism of the reduced variety underlying the scheme Ω_{r} with the opposite ‘cell’ in some Schubert variety X_{v} inside P \GL_{d}, without identifying v = v(r). Indeed, Z is obviously an isomorphism (in fact, a linear map) from Hom to Z(Hom), which is easily seen to equal Y_{v(Hom)}. Then one uses equivariance of Z under appropriate actions of GL to conclude that it takes orbit closures (the reduced varieties underlying quiver loci) to orbit closures (opposite Schubert ‘cells’).

1.4. Quiver polynomials as multidegrees

In this section we define quiver polynomials, our title characters, as multidegrees of quiver loci Ω_{r}, and prove that these agree with the polynomials appearing in [BF99]. For the reader’s convenience, we present the definition of multidegree [BB82, Jos84, BB85, Ros89] and introduce related notions here; those wishing more background and explicit examples can consult [KM03a, Sections 1.2 and 1.7] or [MS04, Chapter 8].

Let 𝕜[f] be a polynomial ring graded by ℤ^{d}. The gradings we consider here are all^{3} positive in the sense of [KM03a, Section 1.7], which implies that the graded pieces of 𝕜[f] have finite dimension as vector spaces. Writing X = x_{1}, …, x_{d} for a basis of ℤ^{d}, each variable f ∈ 𝕜[f] has an ordinary weight deg(f) = u(f) = ∑_{i=1}^{d} u_{i}(f) · x_{i} ∈ ℤ^{d}, and the corresponding exponential weight wt(f) = X^{u(f)} = ∏_{i=1}^{d} x_{i}^{u_{i}(f)}, which is a Laurent monomial in the ring ℤ[x_{1}^{±1}, …, x_{d}^{±1}]. Every finitely generated ℤ^{d}-graded module Γ = ⨁_{u∈ℤ^{d}} Γ_{u} over 𝕜[f] has a finite ℤ^{d}-graded free resolution

E. : 0 ← E_{0} ← E_{1} ← E_{2} ← ⋯  where  E_{i} = ⨁_{j=1}^{β_{i}} 𝕜[f](−t_{ij})

is ℤ^{d}-graded, with the j^{th} summand of E_{i} generated in degree t_{ij} ∈ ℤ^{d}. In this case, the K-polynomial of Γ is

K(Γ; X) = ∑_{i}(−1)^{i} ∑_{j} X^{t_{ij}}.

Given any Laurent monomial X^{u} = x_{1}^{u_{1}} ⋯ x_{d}^{u_{d}}, the rational function ∏_{j=1}^{d}(1 − x_{j})^{u_{j}} can be expanded as a well-defined (that is, convergent in the X -adic topology) formal power series ∏_{j=1}^{d}(1 − u_{j}x_{j} + ⋯) in X. Doing the same for each monomial in an arbitrary Laurent polynomial K(X) results in a power series denoted by K(1 − X). The multidegree of a ℤ^{d}-graded 𝕜[f]-module Γ is

C(Γ; X) = the sum of terms of degree codim(Γ) in K(Γ; 1 − X),

where codim(Γ) = dim(Spec(𝕜[f])) − dim(Γ). If Γ = 𝕜[f]/I is the coordinate ring of a subscheme X ⊆ Spec(𝕜[f]), then we may also write [X]_{ℤ^{d}} or C(X; X) to mean C(Γ; X).

3Actually, a nonpositive grading appears in the proof of Proposition 2.6, but we apply no theorems to it—the nonpositive grading there is only a transition between two positive gradings.

In the context of quiver loci, d = r_{0} + ⋯ + r_{n}, and we write x_{r} for the alphabet X, which is split into a sequence of n + 1 alphabets

(1.12)  x_{r} = x_{r}^{0}, …, x_{r}^{n}

of sizes r_{0}, …, r_{n}. Thus the i^{th} alphabet is

(1.13)  x_{r}^{i} = x_{1}^{i}, …, x_{r_{i}}^{i}.

The coordinate ring 𝕜[Hom] is graded by ℤ^{d}, with the variable f^{i}_{αβ} ∈ 𝕜[Hom] having

ordinary weight deg(f^{i}_{αβ}) = x_{α}^{i−1} − x_{β}^{i}
and exponential weight wt(f^{i}_{αβ}) = x_{α}^{i−1}/x_{β}^{i}

(1.14)

for each i = 1, …, n. Under this grading by ℤ^{d}, the quiver ideal I_{r} is homogeneous.

Definition 1.18. The (ordinary) quiver polynomial is the multidegree

Q_{r}(x − ^{∘}x) = C(Ω_{r}; x_{r})

of the subvariety Ω_{r} inside Hom, under the ℤ^{d}-grading in which deg(f^{i}_{αβ}) = x_{α}^{i−1} − x_{β}^{i}.

For the moment, the argument x−^{∘}x of Q_{r} can be regarded as a formal symbol, denoting that n + 1 alphabets x = x^{0}, …, x^{n} are required as input. These alphabets may without harm be thought of as infinite, as in (1.17) and (1.18), even though all but the finitely many variables in x_{r} are to be ignored. Later, in Section 2.2, we shall define ‘double quiver polynomials’, with arguments x−^{∘}y, indicating two sequences of alphabets as input; then the symbol ^{∘}x will take on additional meaning as the reversed sequence x^{n}, …, x^{0} in Section 2.3.

Our discussion of quiver polynomials will require supersymmetric Schur functions. Given (finite or infinite) alphabets X and Y, define for r ∈ ℕ the homogeneous symmetric function h_{r}(X − Y) in the difference X − Y of alphabets by the generating series

\frac{∏_{y∈Y}(1 − zy)}{∏_{x∈X} (1 − zx)} = ∑_{r≥0} z^{r}h_{r}(X − Y),

in which h_{r}(X − Y) = 0 for r < 0. Then the Schur function s_{λ}(X − Y) in the difference X − Y of alphabets is defined by the Jacobi–Trudi determinant

s_{λ}(X − Y) = det (h_{λ_{i}−i+j}(X − Y)).

The right-hand side is the determinant of an ℓ × ℓ matrix, where ℓ is the number of parts in the partition λ.

Buch and Fulton defined polynomials in [BF99] that also deserve rightly to be called ‘quiver polynomials’ (though they did not use this appellation). Given a sequence of maps between vector bundles E_{0}, …, E_{n} on a scheme, the polynomials appearing in their Main Theorem take as arguments the Chern classes of the bundles. Here, we shall express these polynomials formally as symmetric functions of the Chern roots

(1.15)  x_{r} = {x_{j}^{i} | i = 0, …, n and j = 1, …, r_{i}}

of the dual bundles E_{0}^{∨}, …, E_{n}^{∨}. The Main Theorem of [BF99] is expressed in terms of integers c_{\underline{λ}}(r) called quiver constants, each depending on a rank array r and a sequence \underline{λ} = (λ_{1}, λ_{2}, …, λ_{n}) of n partitions. Let ∑ c_{\underline{λ}}(r)s_{\underline{λ}}(x − ^{∘}x) be the corresponding sum of products

(1.16)  s_{\underline{λ}}(x − ^{∘}x) = ∏_{i=1}^{n} s_{λ_{j}}(x^{i−1} − x^{i})

of Schur functions in differences of consecutive alphabets from the sequence

(1.17)  x = x^{0}, …, x^{n}.

In this notation, each of the alphabets

(1.18)  x^{j} = x_{1}^{j}, x_{2}^{j}, x_{3}^{j}, …

is taken to be infinite, so that s_{\underline{λ}}(x − ^{∘}x) is a power series symmetric in each of n + 1 sets of infinitely many variables. This notation allows us to evaluate s_{\underline{λ}} on finite alphabets of any given size, by using x_{r} as in (1.12) and (1.13). It results in our notation ∑ c_{\underline{λ}}(r)s_{\underline{λ}}(x_{r} − ^{∘}x_{r}) for the polynomials appearing in the Main Theorem of [BF99]. However, since the variables x_{r} in (1.15) are Chern roots of dual bundles, our c_{\underline{λ}}(r) would be denoted by c_{\underline{λ}}′(r) in [BF99], where \underline{λ}′ = (λ′_{1}, …, λ′_{n}) is the sequence of conjugate (i.e., transposed) partitions. This follows because s_{λ}(X − Y) = s_{λ′}((−Y) − (−X)); see also [BF99, Section 4].

Before getting to Theorem 1.20, we must first say more about the geometry underlying multidegrees, which are algebraic substitutes for equivariant cohomology classes of sheaves on torus representations. Here, the torus is a maximal torus T inside GL acting on Hom. We choose T so the bases for V_{0}, …, V_{n} that were fixed in Section 1.1 consist of eigenvectors. Thus T consists of sequences of diagonal matrices, of sizes r_{0}, …, r_{n}. The group ℤ^{d} in this context is the weight lattice of the torus T, and x_{j}^{i} is the weight corresponding to the action of the j^{th} diagonal entry in the i^{th} matrix. The action of T on Hom via the inclusion T ⊂ GL induces the grading (1.14) on 𝕜[Hom].

The geometric data of a torus action on Hom determines an equivariant Chow ring A_{T}^{∗} (Hom). Let us recall briefly the relevant definitions from [EG98] in the context of a linear algebraic group G acting on a smooth scheme M. Let W be a representation of G containing an open subset U ⊂ W on which G acts freely. For any integer k, we can choose W and U so that W ∖ U has codimension at least k + 1 inside W [EG98, Lemma 9 in Section 6], and the construction is independent of our particular choices by [EG98, Definition–Proposition 1 and Proposition 4]. The G-equivariant Chow ring A_{G}^{∗}(M) has degree k piece

A_{G}^{k}(M) = A^{k}((M × U)/G)

equal to the degree k piece of the ordinary Chow ring of the quotient of M × U modulo the diagonal action of G. We have used smoothness of M via [EG98, Corollary 2].

The equivariant Chow rings A_{T}^{∗} (M) for torus actions on vector spaces M all equal the integral symmetric algebra Sym._{ℤ}(𝔱_{ℤ}^{∗}) of the weight lattice 𝔱_{ℤ}^{∗}, independently of M. Thus, once we pick a basis X = x_{1}, …, x_{d} for 𝔱_{ℤ}^{∗}, the class of any T -stable subvariety is uniquely expressed as a polynomial in ℤ[X]. More generally, every ℤ^{d}-graded module Γ (equivalently, T -equivariant sheaf Γ̃) determines a polynomial [cycle(Γ)]_{T}, where the cycle of Γ is the sum of its maximal-dimensional support components with coefficients given by the generic multiplicity of Γ along each.

Proposition 1.19. The multidegree C(Γ; X) of a ℤ^{d}-graded module Γ over 𝕜[f] is the class [cycle(Γ)]_{T} in the equivariant Chow ring A_{T}^{∗} (M) of M = Spec(𝕜[f]).

Proof. The function sending each graded module to its multidegree is characterized uniquely as having the ‘Additivity’, ‘Degeneration’, and ‘Normalization’ properties in [KM03a, Theorem 1.7.1]. Thus it suffices to show that the function sending each module to the equivariant Chow class of its cycle also has these properties. Additivity for Chow classes is by definition. Degeneration is the invariance of Chow classes under rational equivalence of cycles. Normalization, which says that a coordinate subspace L ⊆ M has class equal to the product of the weights of the variables in its defining ideal, follows from [Bri97, Theorem 2.1] and [EG98, Proposition 4] by downward induction on the dimension of L. □

Theorem 1.20. The quiver polynomial Q_{r}(x − ^{∘}x) in Definition 1.18 coincides with the degeneracy locus polynomial ∑ c_{\underline{λ}}(r)s_{\underline{λ}}(x_{r} − ^{∘}x_{r}) in the Main Theorem of [BF99] (although our notation for this polynomial differs from [BF99]; see the lines after (1.18), above).

Proof. Let Gr(r, ℓ) be the product Gr(r_{0}, ℓ) × ⋯ × Gr(r_{n}, ℓ) of Grassmannians of subspaces of dimensions r_{0}, …, r_{n} inside 𝕜^{ℓ} for some large ℓ. This variety Gr(r, ℓ) comes endowed with n + 1 universal vector bundles V_{0}, …, V_{n} of ranks r_{0}, …, r_{n}. Denote by Hom the vector bundle ∏_{j=1}^{n} Hom(V_{i−1}, V_{i}) over Gr(r, ℓ). Pulling back V_{i} to Hom for all i yields vector bundles with a universal (or “tautological”) sequence

Φ̃ : Ṽ_{0} \underrightarrow{Φ̃_{1}} Ṽ_{1} \underrightarrow{Φ̃_{2}} ⋯ \underrightarrow{Φ̃_{n−1}} Ṽ_{n−1} \underrightarrow{Φ̃_{n}} Ṽ_{n}.

of maps between them. Let Ω_{r} ⊆ Hom denote the degeneracy locus of Φ̃ for ranks r. If k = d(r) is the codimension of Ω_{r} from Corollary 1.16, then we can assume ℓ has been chosen so large that the graded component A^{k}(Hom) of the Chow ring of Hom consists of all polynomials of degree k in ℤ[x_{r}] symmetric in each of the n+1 alphabets x_{r}^{i} from (1.13).

The class [Ω_{r}] ∈ A^{k}(Hom) is by definition the polynomial in the Main Theorem of [BF99]. On the other hand, A^{k}(Hom) is by [EG98, Corollary 2] the degree k piece A_{GL}^{k}(Hom) of the GL-equivariant Chow ring of Hom, and the ordinary class [Ω_{r}] ∈ A^{k}(Hom) coincides with the equivariant class [Ω_{r}]_{GL} ∈ A_{GL}^{k}(Hom). The result now follows from Proposition 1.19, given that A_{GL}^{∗}(Hom) consists of the symmetric group invariants inside A_{T}^{∗} (Hom) by [EG98, Proposition 6]. □

Remark 1.21. Thinking of quiver polynomials as equivariant Chow or equivariant cohomology classes makes sense from the topological viewpoint taken originally by Thom [Tho55] in the context of what we now know as the Giambelli–Thom–Porteous formula. In the present context, a quiver of vector bundles on a space Y is induced by a unique homotopy class of maps from Y to the universal bundle Hom_{GL} with fiber Hom on the classifying space BGL of GL. The induced map H_{GL}^{∗}(point) = H^{∗}(BGL) → H^{∗}(Y) on cohomology simply evaluates the universal degeneracy locus polynomial for r, namely the quiver polynomial, on the Chern classes of the vector bundles in the quiver. Replacing the classifying space for GL by the corresponding Artin stack accomplishes the same thing in the Chow ring context instead of cohomology.

Section 2. Double quiver polynomials

2.1. Double Schubert polynomials

Let d be a positive integer, X = (x_{1}, x_{2}, …, x_{d}) and Y = (y_{1}, y_{2}, …, y_{d}) two sets of variables, and w ∈ S_{d} a permutation. The double Schubert polynomial 𝔖_{w}(X − Y) of Lascoux and Schützenberger [LS82] is defined by downward induction on the length of w as follows. For the long permutation w_{0} ∈ S_{d} that reverses the order of 1, …, d, set

𝔖_{w_{0}}(X − Y) = ∏_{i+j≤d}(x_{i} − y_{j}).

If w ≠ w_{0}, then choose q so that w(q) < w(q + 1), and define

𝔖_{w}(X − Y) = ∂_{q}𝔖_{ws_{q}}(X − Y).

The operator ∂_{q} is the q^{th} divided difference acting on the Y variables, so

(2.1)  ∂_{q}P (x_{1}, x_{2}, …) = \frac{P (x_{1}, x_{2}, …) − P (x_{1}, …, x_{q−1}, x_{q+1}, x_{q}, x_{q+2}, …)}{x_{q} − x_{q+1}}

for any polynomial P taking the X variables and possibly other variables for input.

In Section 6 we will require Schubert polynomials for partial permutations w (Definition 1.11). By convention, when we write 𝔖_{w} for a partial permutation w of size k × ℓ, we shall mean 𝔖w̃, where w̃ is a minimal-length completion of w to an actual permutation. Thus w̃ can be be any permutation whose matrix agrees with w in the northwest k × ℓ rectangle, and such that in the columns strictly to the right of ℓ, as well as in the rows strictly below k, the nonzero entries of w̃ proceed from northwest to southeast (no nonzero entry is northeast of another). This convention is consistent with Definition 1.11, in view of the next result, because the matrix Schubert varieties \overline{X}_{w} and \overline{X}w̃ have equal multidegrees, as they are defined by the same equations (though in a priori different polynomial rings).

The following statement appears (in similar language) as [KM03a, Theorem A]; an equivalent result over the complex numbers 𝕜 = ℂ (given the translation between equivariant Chow and equivariant cohomology) is given in [FR02b]. These results are now seen to be essentially equivalent to [Ful92], via equivariant Chow groups as in the argument of Theorem 1.20.

Theorem 2.1. Let w be a partial permutation matrix of size k × ℓ, so \overline{X}_{w} is a subvariety of M_{kℓ} = Spec(𝕜[f]), where f = (f_{αβ}) for α = 1, …, k and β = 1, …, ℓ. If f_{αβ} has ordinary weight deg(f_{αβ}) = x_{α} − y_{β}, then \overline{X}_{w} has multidegree

C(\overline{X}_{w}; X , Y) = 𝔖_{w}(X − Y)

equal to the double Schubert polynomial for w. (We remind the reader that the matrix for a partial permutation w has a 1 in row q and column p if and only if w(q) = p.)

When the partial permutation w is an honest permutation v ∈ S_{d} with d = r_{0} + ⋯ + r_{n}, we often write 𝔖_{v}(X −Y) = 𝔖_{v}(x_{r} −^{∘}y_{r}) and break the variables into sequences of alphabets

(2.2)  x_{r} = x_{r}^{0}, …, x_{r}^{n} and ^{∘}y_{r} = y_{r}^{n}, …, y_{r}^{0},
(2.3)  where x_{r}^{j} = x_{1}^{j}, …, x_{r_{j}}^{j} and y_{r}^{j} = y_{1}^{j}, …, y_{r_{j}}^{j}.

On the other hand, most partial permutations w that occur in the sequel will have size r_{j−1} × r_{j} for some j ∈ {1, …, n}. In that case we will be interested in 𝔖_{w}(x^{j−1} − y^{j}); as with quiver polynomials, we allow the input alphabets x^{j−1} and y^{j} to be infinite here, even though only the finitely many variables in x_{r}^{j−1} and y_{r}^{j} can actually appear.

2.2. Double quiver polynomials

The coordinate ring 𝕜[M_{d}] of the d × d matrices is multigraded by the group ℤ^{2d} = (ℤ^{r_{0}} ⊕ ⋯ ⊕ ℤ^{r_{n}})^{2}, which we take to have basis as in (2.2) and (2.3). In our context, it is most natural to index the variables f_{αβ}^{ij} in the generic d × d matrix in a slightly unusual manner: f_{αβ}^{ij} ∈ 𝕜[M_{d}] occupies the spot in row α and column β within the rectangle at the intersection of the i^{th} block row and the j^{th} block column, where i, j = 0, …, n and we label block columns starting from the right. Set the ordinary weight of f_{αβ}^{ij} equal to

deg(f_{αβ}^{ij} ) = x_{α}^{i} − y_{β}^{j}

More pictorially, label the rows of the d × d grid with the x basis vectors in the order they are given, from top to bottom, and similarly label the columns with ^{∘}y_{r} = y_{r}^{n}, …, y_{r}^{0}, from left to right (go left to right within each alphabet). Under our multigrading, the ordinary weight of a variable is its row basis vector minus its column basis vector. Therefore:

• Rows in the j^{th} block from the top are labeled x_{1}^{j}, …, x_{r}^{j} for r = r_{jj}, starting with x_{1}^{j} in the highest row of that block.

• Columns in the j^{th} block from the right are labeled y_{1}^{j}, …, y_{r}^{j} for r = r_{jj}, starting

with y_{1}^{j} above the leftmost column of that block. For notational clarity in examples, it is usually convenient to rename the alphabets

x^{0}, x^{1}, x^{2}, … using distinct Latin names, such as

x^{0} = a = a_{1}, a_{2}, a_{3}, … and x^{1} = b = b_{1}, b_{2}, b_{3}, … and x^{2} = c = c_{1}, c_{2}, c_{3}, …

and so on, rather than upper indices. Then, we rename the alphabets y^{0}, y^{1}, y^{2}, … as

y^{0} = ȧ = ȧ_{1}, ȧ_{2}, ȧ_{3}, … and y^{1} = ḃ = ḃ_{1}, ḃ_{2}, ḃ_{3}, … and y^{2} = ċ = ċ_{1}, ċ_{2}, ċ_{3}, …

and so on, the same as the x’s but with dots on top.

All of the notation above should be made clearer by the following example.

Example 2.2. Consider quivers on the vector spaces V = (V_{0}, V_{1}, V_{2}) with dimensions (r_{0}, r_{1}, r_{2}) = (2, 3, 1). The coordinate ring 𝕜[M_{d}] has variables f_{αβ}^{ij} as they appear in the matrices below (the f variables are the same in both):

\begin{aligned}& y_{1}^{2} & y_{1}^{1} & y_{2}^{1} & y_{3}^{1} & y_{1}^{0} & y_{2}^{0} \\ x_{1}^{0} & f_{11}^{02} & f_{11}^{01} & f_{12}^{01} & f_{13}^{01} & f_{11}^{00} & f_{12}^{00} \\ x_{2}^{0} & f_{21}^{02} & f_{21}^{01} & f_{22}^{01} & f_{23}^{01} & f_{21}^{00} & f_{22}^{00} \\ x_{1}^{1} & f_{11}^{12} & f_{11}^{11} & f_{12}^{11} & f_{13}^{11} & f_{11}^{10} & f_{12}^{10} \\ x_{2}^{1} & f_{21}^{12} & f_{21}^{11} & f_{22}^{11} & f_{23}^{11} & f_{21}^{10} & f_{22}^{10} \\ x_{3}^{1} & f_{31}^{12} & f_{31}^{11} & f_{32}^{11} & f_{33}^{11} & f_{31}^{10} & f_{32}^{10} \\ x_{1}^{2} & f_{11}^{22} & f_{11}^{21} & f_{12}^{21} & f_{13}^{21} & f_{11}^{20} & f_{12}^{20} \end{aligned}  = \begin{aligned}& ċ_{1} & ḃ_{1} & ḃ_{2} & ḃ_{3} & ȧ_{1} & ȧ_{2} \\ a_{1} & f_{11}^{02} & f_{11}^{01} & f_{12}^{01} & f_{13}^{01} & f_{11}^{00} & f_{12}^{00} \\ a_{2} & f_{21}^{02} & f_{21}^{01} & f_{22}^{01} & f_{23}^{01} & f_{21}^{00} & f_{22}^{00} \\ b_{1} & f_{11}^{12} & f_{11}^{11} & f_{12}^{11} & f_{13}^{11} & f_{11}^{10} & f_{12}^{10} \\ b_{2} & f_{21}^{12} & f_{21}^{11} & f_{22}^{11} & f_{23}^{11} & f_{21}^{10} & f_{22}^{10} \\ b_{3} & f_{31}^{12} & f_{31}^{11} & f_{32}^{11} & f_{33}^{11} & f_{31}^{10} & f_{32}^{10} \\ c_{1} & f_{11}^{22} & f_{11}^{21} & f_{12}^{21} & f_{13}^{21} & f_{11}^{20} & f_{12}^{20} \end{aligned}

The ordinary weight of each f variable equals its row label minus its column label, and its exponential weight is the ratio instead of the difference. For example, the variable f_{23}^{01} has ordinary weight x_{2}^{0} − y_{3}^{1} = a_{2} − ḃ_{3} and exponential weight x_{2}^{0}/y_{3}^{1} = a_{2}/ḃ_{3}. □

The coordinate ring 𝕜[Y_{0}] of the variety Y_{0} from Section 1.3 is not naturally multigraded by all of ℤ^{2d}, but only by ℤ^{d}, with the variable f_{αβ}^{ij} ∈ 𝕜[Y_{0}] having ordinary weight x_{α}^{i} − x_{β}^{j}. This convention is consistent with the multigrading of 𝕜[Hom] in (1.14) under the homomorphism to 𝕜[Y_{0}] induced by the Zelevinsky map, since the variable f_{αβ}^{i−1,i} ∈ 𝕜[Y_{0}] maps to f^{i}_{αβ} ∈ 𝕜[Hom], and their ordinary weights x_{α}^{i−1} − x_{β}^{i} agree. Our ℤ^{d}-grading of 𝕜[Y_{0}] is positive, so every quotient of 𝕜[Y_{0}] has a Hilbert series.

Example 2.3. In the situation of Example 2.2, the coordinate ring 𝕜[Y_{0}] has only the variables f_{αβ}^{ij} that appear in the matrices below:

\begin{aligned}& x_{1}^{2} & x_{1}^{1} & x_{2}^{1} & x_{3}^{1} & x_{1}^{0} & x_{2}^{0} \\ x_{1}^{0} & f_{11}^{02} & f_{11}^{01} & f_{12}^{01} & f_{13}^{01} & 1 \\ x_{2}^{0} & f_{21}^{02} & f_{21}^{01} & f_{22}^{01} & f_{23}^{01} & & 1 \\ x_{1}^{1} & f_{11}^{12} & 1 \\ x_{2}^{1} & f_{21}^{12} & & 1 \\ x_{3}^{1} & f_{31}^{12} & & & 1 \\ x_{1}^{2} & 1 \end{aligned}  = \begin{aligned}& c_{1} & b_{1} & b_{2} & b_{3} & a_{1} & a_{2} \\ a_{1} & f_{11}^{02} & f_{11}^{01} & f_{12}^{01} & f_{13}^{01} & 1 \\ a_{2} & f_{21}^{02} & f_{21}^{01} & f_{22}^{01} & f_{23}^{01} & & 1 \\ b_{1} & f_{11}^{12} & 1 \\ b_{2} & f_{21}^{12} & & 1 \\ b_{3} & f_{31}^{12} & & & 1 \\ c_{1} & 1 \end{aligned}

In this case, the variable f_{23}^{01} has ordinary weight x_{2}^{0} − x_{3}^{1} = a_{2} − b_{3} and exponential weight x_{2}^{0}/x_{3}^{1} = a_{2}/b_{3}. □

The multigrading giving rise to the ordinary quiver polynomials in Section 1 is derived from the action of GL on Hom. In the current context of GL_{d} acting on M_{d}, we identify GL = ∏_{i=0}^{n} GL(V_{i}) as the Levi factor L, consisting of block diagonal matrices inside the parabolic subgroup P ⊂ GL_{d}, by sticking GL(V_{i}) in the i^{th} diagonal block. This identifies the torus T of diagonal matrices in L with the torus T from Section 1.4.

Lemma 2.4. The matrix Schubert variety \overline{X}_{v(r)} is the closure inside M_{d} of the variety P Z(Ω_{r}), by which we mean the image of P × Z(Ω_{r}) under multiplication.

Proof. In general, if X_{v} is a Schubert subvariety of P \GL_{d}, and X̃_{v} is the preimage of X_{v} inside GL_{d} (under projection modulo P), then \overline{X}_{v} equals the closure of X̃_{v} inside M_{d}. Theorem 1.14 implies that P Z(Ω_{r}) is dense inside the preimage of X_{v(r)}. □

The quiver locus Ω_{r} has only one copy of the torus T ⊂ GL acting on it, as does its Zelevinsky image Z(Ω_{r}) (conjugation by T ⊂ L). Multiplying by P on the left as in Lemma 2.4 frees up the left and right-hand tori (and even the left and right-hand Levi factors), allowing them to act independently. Consequently, we can define a doubly equivariant cohomology class on the smeared Zelevinsky image \overline{X}_{v(r)} of Ω_{r}. In terms of multidegrees, we can now use the two sequences x and ^{∘}y of alphabets instead of only the x alphabets in ordinary quiver polynomials.

Definition 2.5. The double quiver polynomial is the ratio

Q_{r}(x − ^{∘}y) = \frac{𝔖_{v(r)}(x_{r} − ^{∘}y_{r})}{𝔖_{v(Hom)}(x_{r} − ^{∘}y_{r})}

of double Schubert polynomials in the concatenations of the two sequences of finite alphabets described in (2.2) and (2.3).

For the purposes of results such as Theorem 1.20 as well as Corollary 4.9 and Corollary 6.23, it is useful to think of the argument x − ^{∘}y of the double quiver polynomial on the left side of Definition 2.5 as two sequences x and ^{∘}y of alphabets, even though the double Schubert polynomials on the right side more naturally take the argument x_{r} − ^{∘}y_{r} thought of simply as a pair of alphabets x_{r} and ^{∘}y_{r}. In addition, like ordinary quiver polynomials, it does no harm to think of double quiver polynomials as being evaluated on finite sequences of infinite alphabets, with all but the finitely many x_{r} and ^{∘}y_{r} variables ignored.

The denominator 𝔖_{v(Hom)}(x_{r} −^{∘}y_{r}) should be regarded as a fudge factor, being simply the product of all weights (x_{∗} −y_{∗}) of variables lying strictly above the block superantidiagonal; see also Remark 2.10. These variables lie in locations corresponding to ∗ entries in the diagram of every Zelevinsky permutation, so 𝔖_{v(Hom)} obviously divides 𝔖_{v(r)} (see Theorem 5.3).

The simple relation between double and ordinary quiver polynomials, to be presented in Theorem 2.9, justifies the notation Q_{r}(x − ^{∘}x) for the ordinary case.

2.3. Ratio formula for quiver polynomials

Our goal in this section is to express quiver polynomials as the specializations of double quiver polynomials obtained by setting y^{j} = x^{j} for all j.

More generally, we shall express the quiver K-polynomial as a ratio of corresponding specialized double Grothendieck polynomials. These are usually defined by ‘isobaric divided difference operators’ (or ‘Demazure operators’), which we review in Section 7.1.

However, since Grothendieck polynomials are tangential to the focus here, we do not review the background, which can be found in [LS82, Las90] or [KM03a], the latter being well-suited to the current applications. Readers unfamiliar with Grothendieck polynomials will lose nothing in the present context by taking the first sentence of Proposition 2.6 as a definition.

Double Grothendieck polynomials G_{v}(X /Y) for permutations v ∈ S_{d} take as arguments two alphabets X and Y, each having size d. For purposes relating to quiver loci, X and Y are equated with sequences x_{r} and y_{r} of alphabets as in (2.2) and (2.3). Write

∘x_{r} = x_{r}^{n}, …, x_{r}^{0}

to mean the alphabet obtained by block reversing the finite list x_{r} of alphabets from (2.2). It will arise when calculating the K-polynomials of opposite ‘cells’ in Schubert subvarieties of partial flag manifolds P \GL_{d}, as we shall see shortly. Note that in the case of full flags, where P = B is the Borel subgroup of lower-triangular matrices in GL_{d}, each alphabet in the list x_{r} consists of just one variable (as opposed to there being only one alphabet in the list), so the reversed sequence ^{∘}x_{r} is really just a reversed alphabet in that case.

For notation, let K_{M}(\overline{X}; X , Y) denote the K-polynomial of a multigraded subvariety \overline{X} ⊆ M_{d} under the ℤ^{2d}-grading from Section 2.2, and let K_{Y} (Z; X) denote the K-polynomial of a multigraded subvariety Z ⊆ Y_{0} under the ℤ^{d}-grading there.

Proposition 2.6. The K-polynomial K_{M}(\overline{X}_{v}; X , Y) of a matrix Schubert variety \overline{X}_{v} ⊆ M_{d} is the double Grothendieck polynomial G_{v}(X /Y). Intersecting \overline{X}_{v(r)} with Y_{0} yields the variety Z(Ω_{r}), whose K-polynomial K_{Y} (Z(Ω_{r}); x_{r}) is the specialization G_{v(r)}(x_{r}/^{∘}x_{r}).

Proof. The first sentence follows from [KM03a, Theorem A], one of the main results there. For the second sentence, observe that Z(Ω_{r}) has the same codimension inside Y_{0} as does \overline{X}_{v(r)} inside M_{d}. Hence the equations defining Y_{0} as a subvariety of M_{d} (namely f − 1 for diagonal f in antidiagonal blocks and f − 0 for the other variables in or below the main block antidiagonal) form a regular sequence on \overline{X}_{v(r)}. Coarsening the ℤ^{2d}-grading on 𝕜[M_{d}] to the grading by ℤ^{d} in which f_{αβ}^{ij} has ordinary weight x_{α}^{i} − x_{β}^{j}, by setting y_{β}^{j} = x_{β}^{j}, makes the equations defining Y_{0} become homogeneous, because the variables set equal to 1 have weight zero. Therefore, if F. is any ℤ^{2d}-graded free resolution of 𝕜[\overline{X}_{v(r)}] over 𝕜[M_{d}], then F. ⊗_{𝕜[M_{d}]} 𝕜[Y_{0}] is a ℤ^{d}-graded free resolution of 𝕜[Z(Ω_{r})] over 𝕜[Y_{0}]. It follows that K_{M}(\overline{X}_{v(r)}; x_{r}, ^{∘}x_{r}) = K_{Y} (Z(Ω_{r}); x_{r}), because the relevant ℤ^{d}-graded Betti numbers coincide. Applying the first sentence of the Proposition and substituting y_{β}^{j} = x_{β}^{j} now proves the second sentence. □

Theorem 2.7. The K-polynomial K_{Hom}(Ω_{r}; x_{r}) of Ω_{r} inside Hom is the ratio

K_{Hom}(Ω_{r}; x_{r}) = \frac{G_{v(r)}(x_{r}/^{∘}x_{r})}{G_{v(Hom)}(x_{r}/x^{∘}_{r})}

of specialized double Grothendieck polynomials for v(r) and v(Hom).

Proof. The equality H(Ω_{r}; x_{r}) = K_{Hom}(Ω_{r}; x_{r})H(Hom; x_{r}) of Hilbert series (which are welldefined by positivity of the grading of 𝕜[Hom] by ℤ^{d}) follows from the definition of Kpolynomial. For the same reason, we have H(Hom; x_{r}) = K_{Y} (Z(Hom); x_{r})H(Y_{0}; x_{r}) and also H(Ω_{r}; x_{r}) = K_{Y} (Z(Ω_{r}); x_{r})H(Y_{0}; x_{r}). Therefore

K_{Y} (Z(Ω_{r}); x_{r})H(Y_{0}; x_{r}) = K_{Hom}(Ω_{r}; x_{r})K_{Y} (Z(Hom); x_{r})H(Y_{0}; x_{r}).

Canceling the factors of H(Y_{0}; x_{r}) and dividing by K_{Y} (Z(Hom); x_{r}) yields

K_{Hom}(Ω_{r}; x_{r}) = \frac{K_{Y} (Z(Ω_{r}); x_{r})}{K_{Y} (Z(Hom); x_{r})}.

Now apply Theorem 1.14 and Proposition 2.6 to complete the proof. □

Remark 2.8. Here is the geometric content of Proposition 2.6. The torus T × T of dimension 2d acts on M_{d}, the left factor by multiplication on the left, and the right factor by inverse multiplication on the right. Viewing G_{v}(x/y) as an equivariant class for this action of T × T, specializing the y variables to ^{∘}x does not arise by restricting the T × T action to some subtorus. Rather, it uses a different d-dimensional torus T entirely. The main point is that the opposite big cell is an orbit for an opposite parabolic group, whose torus has been conjugated by the identity-in-each-antidiagonal-block permutation w_{0}w_{0} from the usual one.

In more detail, the subvariety Y_{0} ⊂ GL_{d} is not stable under the usual action of T × T above, because we need to preserve the 1s in antidiagonal blocks. Instead, it is stable under the action of the diagonal torus T inside the copy of T × T whose left factor acts by usual multiplication on the left, but where a diagonal matrix τ in the right factor acts as right multiplication by w_{0}w_{0}τ^{−1}w_{0}w_{0}. Thus the diagonal T acts with the weight on the variable at position (α, β) in block row i and block column j (from the right) being x_{α}^{i} − x_{β}^{j}.

As a reality check, note that this action of the right-hand T factor is indeed the restriction of the “opposite” parabolic action on P \GL_{d} to its torus. Indeed, our “usual” parabolic acting on P \GL_{d} is the block upper-triangular P_{+} ⊂ GL_{d} acting by inverse multiplication on the right (we want P × P_{+} to act via a left group action), so the opposite parabolic is w_{0}w_{0}P_{+}w_{0}w_{0} acting by inverse multiplication on the right.

One might observe that the torus in w_{0}w_{0}P_{+}w_{0}w_{0} acts on P \GL_{d} simply by inverse right multiplication, not by twisted conjugation. All this means is that multiplying by a diagonal matrix w_{0}w_{0}τ^{−1}w_{0}w_{0} moves the cosets in P \GL_{d} correctly. However, the subvariety Y_{0} ⊂ GL_{d} consists of specific coset representatives for points in the opposite big cell of P \GL_{d}, and we need to know which coset representative corresponds to the image of a matrix after right multiplication by w_{0}w_{0}τ^{−1}w_{0}w_{0}. For this, we have to left-multiply by τ, bringing the antidiagonal blocks back to a list of identity matrices without altering the coset.

Next comes what we have been after from the beginning, the consequence of Theorem 2.7 on multidegrees (or equivariant cohomology or Chow groups, by Proposition 1.19).

Theorem 2.9. The quiver polynomial Q_{r}(x − ^{∘}x) for a rank array r equals the ratio

Q_{r}(x − ^{∘}x) = \frac{𝔖_{v(r)}(x_{r} − ^{∘}x_{r})}{𝔖_{v(Hom)}(x_{r} − x^{∘}_{r})}

of specialized double Schubert polynomials. In other words, the ordinary quiver polynomial Q_{r}(x − ^{∘}x) is the ^{∘}y = ^{∘}x specialization of the double quiver polynomial Q_{r}(x − ^{∘}y).

Proof. Clear denominators in Theorem 2.7, substitute 1 − x for every occurrence of each variable x, take lowest degree terms, and divide the result by 𝔖_{v(Hom)}(x_{r} − ^{∘}x_{r}). Note that 𝔖_{v(Hom)}(x_{r} − ^{∘}x_{r}) is nonzero, being simply the product of the ℤ^{d}-graded weights of the variables in 𝕜[Y_{0}] strictly above the block superantidiagonal. □

Remark 2.10. The proofs of both Proposition 2.6 and Theorem 2.7 could as well have been presented entirely in geometric language, referring to pullbacks and pushforwards in equivariant K-theory rather than ratios of numerators of Hilbert series. In this geometric language, the denominator in Theorem 2.9 results via the push-pull formula from the equivariant Euler class of the normal bundle to Z(Hom) in Y_{0}, or equivalently, the equivariant Euler class of the normal bundle to \overline{X}_{v(Hom)} in M_{d}.

In view of Theorem 2.9, we consider Definition 2.5 to be our first positive formula for double quiver polynomials, since it expresses them in terms of known polynomials (Schubert polynomials) that depend on combinatorial data (the Zelevinsky permutation) associated to given ranks r. This is not so bad: we could have (without motivation) defined double quiver polynomials starting from the Buch–Fulton formula [BF99] by using independent dual bundles in the second half of each argument in their Schur functions, and then Definition 2.5 would be a theorem (but not one in this paper; see Remark 6.25). As we have chosen to present things, the double version of the Buch–Fulton formula will come out as a theorem instead, in Theorems 7.10, 7.21, and 8.22; see also Corollary 8.23.

Section 3. Lacing diagrams

3.1. Geometry of lacing diagrams

In this section we use diagrams derived from those in [AD80] to describe the orbits of certain groups related to GL. Recall that we have fixed bases for the vector spaces V_{0}, …, V_{n}, and this essentially amounted to fixing a maximal torus T inside the group GL acting on V. Suppose that, in this basis, the quiver representation ϕ ∈ Hom is represented by a sequence w = (w_{1}, …, w_{n}) of n partial permutation matrices with respect to the above bases. We shall freely identify ϕ with w.

Each list w = (w_{1}, …, w_{n}) ∈ Hom of partial permutation matrices can be represented by a (nonembedded) graph in the plane called a lacing diagram. The vertex set of the graph consists of r_{i} bottom-justified dots in column i for i = 0, …, n, with an edge connecting the dot at height α (from the bottom) in column i − 1 with the dot at height β in column i if and only if the entry of w_{i} at (α, β) is 1. A lace is a connected component of a lacing diagram. An (i, j)-lace is one that starts in column i and ends in column j. We shall also identify w ∈ Hom with its associated lacing diagram.

Example 3.1. Here is a lacing diagram for the dimension vector (2, 3, 4, 3) and its corresponding list of partial permutation matrices.

⟷ ([\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}] , [\begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}] , [\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}])

This lacing diagram has rank array r and lace array s from Example 1.5. □

Lemma 3.2. Fix a rank array r that can occur for V, and let s be the lace array related to r by (1.2). For a lacing diagram w ∈ Hom, the following are equivalent:

1. w has rank array r.

2. The quiver locus Ω_{r} is the closure \overline{GL · w} of the orbit of GL through w.

3. s_{ij} is the number of (i, j)-laces in w for all 0 ≤ i ≤ j ≤ n.

Proof. This is a consequence of Proposition 1.2 plus Lemma 1.3 and its proof. □

Lemma 3.2 identifies orbits indexed by a collection of lacing diagrams. It will be important for us to consider another group action, for which each lacing diagram uniquely determines its own orbit. As a starting point, we have the following standard result; see [Ful92] or [KM03a]. As a matter of notation, we write w′ ≤ w for partial permutation matrices w and w′ of the same shape if their associated permutations satisfy w′ ≤ w̃ in Bruhat order.

Lemma 3.3. Let the product B_{−}(k) × B_{+}(ℓ) of lower- and upper-triangular Borel groups inside GL_{k} × GL_{ℓ} act on matrices Z ∈ M_{kℓ} by (b_{−}, b_{+}) · Z = b_{−}Zb_{+}^{−1}. The matrix Schubert variety \overline{X}_{w} for a partial permutation matrix w is the closure in M_{kℓ} of the orbit of B_{−}(k) × B_{+}(ℓ) through w. Moreover, \overline{X}_{w′} ⊂ \overline{X}_{w} if and only if w′ ≤ w.

The observation in Lemma 3.3 easily extends to sequences of partial permutations—that is, lacing diagrams w = (w_{1}, …, w_{n})—using the subgroup

B_{+} × B_{−} = ∏_{i=0}^{n} B_{+}(V_{i}) × B_{−}(V_{i})

of the group GL^{2} from (1.1), where B_{+}(V_{i}) and B_{−}(V_{i}) are the upper and lower triangular Borel groups in GL(V_{i}) = GL_{r_{i}}, respectively. The action of B_{+} × B_{−} on Hom is inherited from the GL^{2} action after (1.1). We also need to consider the subgroup

(3.1)  B_{+} ×_{T} B_{−} = ∏_{i=0}^{n} B_{+}(V_{i}) ×_{T (V_{i})} B_{−}(V_{i}) ⊂ B_{+} × B_{−},

where B_{+}(V_{i}) ×_{T (V_{i})} B_{−}(V_{i}) consists of those pairs (b_{+}, b_{−}) ∈ B_{+}(V_{i}) × B_{−}(V_{i}) in which the diagonals of b_{+} and b_{−} are equal.

Proposition 3.4. The groups B_{+} ×_{T} B_{−} and B_{+} × B_{−} have the same orbits on Hom, and each orbit contains a unique lacing diagram w. The closure in Hom of the orbit through w is the product

O(w) = ∏_{j=1}^{n} \overline{X}_{w_{j}}

of matrix Schubert varieties, with \overline{X}_{w_{j}} lying inside the factor Hom(V_{j−1}, V_{j}) of Hom.

Proof. By Lemma 3.3 the action of B_{+} × B_{−} on Hom has orbit representatives given by lacing diagrams. Hence it suffices to show that every orbit of the subgroup B_{+} ×_{T} B_{−} ⊂ B_{+} × B_{−} acting on Hom contains a lacing diagram.

Note that B_{+} ×_{T} B_{−} = T · U for the maximal torus T ⊂ GL (diagonally embedded in GL^{2}) and the unipotent group U = ∏_{i} U_{+}(V_{i}) × U_{−}(V_{i}), where U_{+}(V_{i}) ⊂ B_{+}(V_{i}) and U_{−}(V_{i}) ⊂ B_{−}(V_{i}) are the unipotent radicals (unitriangular matrices). Acting on the factor Hom(V_{i−1}, V_{i}) by the group U_{−}(V_{i−1}) × U_{+}(V_{i}) via the action of GL^{2} after (1.1), any matrix can be transformed into a monomial matrix (one having at most one nonzero element in each row and each column) by performing downward row elimination and rightward column elimination with no scaling of the pivot elements. Thus each orbit of U in Hom contains a list of monomial matrices ϕ = (ϕ_{1}, …, ϕ_{n}).

Let us view a list of monomial matrices as a lacing diagram whose edges have been labeled with nonzero elements of 𝕜. The underlying lacing diagram w is determined by the list of partial permutation matrices obtained by changing each nonzero entry of ϕ to a 1, and if the coordinate f^{i}_{αβ} is nonzero on ϕ, then its value f^{i}_{αβ}(ϕ) is the label on the edge connecting the height α vertex in column i − 1 to the height β vertex in column i. It now suffices to find τ ∈ T with τϕ = w. We construct τ by setting the j^{th} diagonal entry τ_{i}^{j} in its i^{th} factor τ_{i} to be the product of the edge labels that are to the left of column i on the lace in ϕ containing the height j vertex in column i. □

The entry (i, j) of a partial permutation matrix is called an inversion if there is no 1 due north or due west of it, i.e., there is no 1 in position (i′, j) for i′ < i or (i, j′) for j′ < j. Define the length ℓ(w) of a partial permutation matrix w to be the number of inversions of w. Equivalently, this is the length of any permutation of minimal length whose matrix contains w as its northwest corner. Given a list w = (w_{1}, …, w_{n}) of partial permutation matrices, define the length of w to be ℓ(w) = ∑_{j=1}^{n} ℓ(w_{j}).

Definition 3.5. Denote by W (r) the subset of lacing diagrams in Hom with rank array r and length d(r) = codim(Ω_{r}) from Corollary 1.16.

Example 3.6. For r as in Example 1.5, the set W (r) is given by

The codimension of this quiver locus Ω_{r} is 7, which will be reflected in the number of ‘virtual crossings’ in these diagrams, as defined in this next section. □

3.2. Minimum length lacing diagrams

In Theorem 3.8 of this section we give a combinatorial characterization of the lacing diagrams W (r) from Definition 3.5.

For partial permutation matrix lists (lacing diagrams) w and w′, write w′ ≤ w if w′_{j} ≤ w_{j} in Bruhat order for all 1 ≤ j ≤ n.

To compute the length of a lacing diagram we introduce the notion of a crossing. The vertex at height i (from the bottom) in column k is written as the ordered pair (i, k). For the purposes of the following definition one may imagine that every (a, b)-lace is extended so that its left endpoint is connected by an invisible edge to an invisible vertex (∞, a−1) and its right endpoint is connected by an invisible edge to an invisible vertex (∞, b + 1). Two laces in a lacing diagram are said to cross if one contains an edge of the form (i, k − 1) → (j, k) and the other an edge of the form (i′, k − 1) → (j′, k) such that i > i′ and j < j′. Such a crossing shall be labeled by the triple (j, j′, k). Looking at these two edges, this crossing is classified as follows.

Call the crossing ordinary if neither edge is invisible; left-virtual if the edge touching (j, k) is invisible but the other edge is not; right-virtual if the edge touching (j′, k) is invisible but the other is not; and doubly virtual if both edges are invisible. Two laces have a doubly virtual crossing (necessarily unique) if and only if one is an (a, b)-lace and the other a (b + 1, c)-lace for some a ≤ b < c.

For the next proposition and theorem, a crossing refers to any of the above four kinds of crossings.

Proposition 3.7. The number of crossings in any lacing diagram w ∈ Hom is ℓ(w).

Proof. Fix 1 ≤ k ≤ n. The inversions (i, j) of w_{k} come in four flavors:

1. w_{k} has a 1 in positions (i, j′) and (i′, j) with j′ > j and i′ > i.

2. w_{k} has a 1 in position (i, j′) with j′ > j but no 1 in the j^{th} column.

3. w_{k} has a 1 in position (i′, j) with i′ > i but no 1 in the i^{th} row.

4. w_{k} has no 1 in either the i^{th} row or j^{th} column. Consider the two laces in w containing the vertices (i, k − 1) and (j, k) respectively; they are distinct since (i, j) is an inversion of w_{k}. Then the four kinds of inversions correspond respectively to crossings (j, j′, k) of these two laces that are ordinary, left-virtual, rightvirtual, and doubly virtual respectively. □

It is a bit difficult to use this proposition to identify by eye the lacing diagrams in W (r), because it is hard to make an accurate count of virtual crossings. The following theorem gives a more easily checked criterion (though it still involves virtual crossings).

Theorem 3.8. Suppose the lacing diagram w has rank array r. Then ℓ(w) ≥ d(r), with equality if and only if

(C1) no pair of laces in w crosses more than once; and

(C2) two laces in w have no crossings if both start or both end in the same column.

Theorem 3.8 follows immediately from Proposition 3.7 and the two following results, which may be of independent interest.

Proposition 3.9. Suppose the lacing diagram w has rank array r but fails to satisfy (C1) or (C2). Then there is a lacing diagram w′ with rank array r such that w′ < w.

Proof. Suppose first that w has a pair of laces L and L′ that cross at least twice. Then undoing adjacent crossings in these two laces produces the desired lacing diagram w′. More precisely, consider a pair of crossings (j, j′, k) and (j_{1}, j′_{1}, k_{1}) of L and L′ with k < k_{1} and k_{1} − k minimal. Let w′ be the lacing diagram obtained from w by changing the vertex sets of L and L′ by trading vertices in columns c such that k ≤ c < k_{1} (and leaving all other laces unchanged). It is straightforward to verify that w′ has rank array r, that w′_{k} < w_{k}, that w′_{k_{1}} < w_{k_{1}}, and that w′_{c} = w_{c} for c ∉ {k, k_{1}}.

Suppose w has an (a, b)-lace L and an (a, b′)-lace L′ that cross (the case that two crossing laces end in the same column is entirely similar). Let (j, j′, k) be the crossing of L and L′ with k minimal. As before we may change the vertex sets of L and L′ by trading vertices in columns c such that a ≤ c < k, resulting in the lacing diagram w′. Then w′ has rank array r, with w′_{k} < w_{k}, and w′_{c} = w_{c} for a ≤ c < k. □

Proposition 3.10. If w has rank array r then w has at least d(r) crossings. Moreover if w satisfies (C1) and (C2) then w has exactly d(r) crossings.

Proof. Consider any (a, b)-lace L and (a′, b′)-lace L′ in w such that a < a′ and b < b′. It is easy to check that the laces cross at least once if and only if b ≥ a′ − 1. From this and Corollary 1.16 it follows that w has at least d(r)-crossings. Suppose w has rank array r and satisfies (C1) and (C2). Let L be an (a, b) lace and L′ an (a′, b′)-lace with a ≤ a′. If a = a′ the laces cannot cross by (C2), so suppose a < a′. If b > b′ then the laces cannot cross, for if they did then they would have to cross twice, violating (C1). If b = b′ then again by (C2) the laces cannot cross. Finally, for b < b′, the laces cross at least once and only if b ≥ a′ − 1, and they cannot cross twice by (C1). This gives precisely the desired d(r) crossings. □

Section 4. Degeneration of quiver loci

4.1. Orbit degenerations

Tracing the degeneration of a group acting on a space can allow one to degenerate the orbits of the group. More precisely, the next result says that under appropriate conditions, an action of a one-dimensional flat family \overline{G} of groups on the general fiber X of a flat family \overline{X} extends to a \overline{G}-action on all of \overline{X}. This implies, in particular, that the special fiber of \overline{G} acts on the special fiber of \overline{X}.

Proposition 4.1. Let the group scheme Γ_{0} act on affine n-space 𝔸^{n} (over the integers, say). Fix a regular scheme \overline{A} of dimension 1 along with a closed point a ∈ \overline{A}, and consider the family \overline{V} = 𝔸^{n} × \overline{A}. Suppose \overline{G} is a flat closed subgroup scheme of \overline{Γ} ≅ Γ_{0} × \overline{A} over \overline{A}. Let V, G, and Γ denote the restrictions of \overline{V}, \overline{G}, and \overline{Γ} to schemes over A = \overline{A} ∖ a. If X ⊆ V is a G-subscheme flat over A, then there is a unique equivariant completion of X to a closed \overline{G}-subscheme \overline{X} ⊆ \overline{V} flat over \overline{A}.

Proof. The flat family X ⊆ V has a unique extension to a flat family \overline{X} whose total space is closed inside \overline{V}; this is standard, taking \overline{X} to be the closure of \overline{X} inside \overline{V}. (The ideal sheaf I\overline{X} is thus I_{X} ∩ O\overline{V}, the intersection taking place inside O_{V}.) It remains to show that the map \overline{G} ×\overline{A} \overline{X} → \overline{V} has image contained in \overline{X} (more formally, the map factors through the inclusion \overline{X} ↪ \overline{V}).

The product G ×_{A} X is a closed subscheme of Γ ×_{A} V, which is itself an open subscheme of \overline{Γ} ×\overline{A} \overline{V}. As such we can take the closure \overline{G ×_{A} X} inside \overline{Γ} ×\overline{A} \overline{V}. The composite map

\overline{G ×_{A} X} ⟶ \overline{Γ} ×\overline{A} \overline{V} ⟶ \overline{V}

lands inside \overline{X}, because (i) removing all the overbars in the above composite map yields a map whose image is precisely the G-scheme X, and (ii) \overline{X} is closed in \overline{V}. It suffices to show now that \overline{G ×_{A} X} = \overline{G} ×\overline{A} \overline{X} as subschemes of \overline{Γ} ×\overline{A} \overline{V}.

Clearly \overline{G ×_{A} X} ⊆ \overline{G}×\overline{A}\overline{X}, because \overline{G}×\overline{A}\overline{X} is closed in \overline{V} and contains G×_{A}X. Moreover, \overline{G} ×\overline{A} \overline{X} is flat over \overline{A} (because both \overline{G} and \overline{X} are) and agrees with G ×_{A} X over A. Thus \overline{G} ×\overline{A} \overline{X} and \overline{G ×_{A} X} both equal the unique extension of the family G ×_{A} X → A to a flat family over \overline{A} that is closed inside \overline{Γ} ×\overline{A} \overline{V}. □

In our case, the group degeneration is a group scheme GL over 𝔸^{1} (parametrized by the coordinate t on 𝔸^{1}). GL is defined as follows via Gröbner degeneration of the diagonal embedding

Δ : GL ↪ GL^{2}.

Let g_{i} = (g_{i}^{αβ})_{α,β=1}^{r_{i}} be coordinates on GL(V_{i}) for each index i, so the coordinate ring

𝕜[GL(V_{i})^{2}] = 𝕜[\overleftarrow{g_{i}}, \overrightarrow{g_{i}}][\overleftarrow{det}^{−1}, \overrightarrow{det}^{−1}]

is the polynomial ring with the determinants of the left and right pointing variables inverted. The ideal of the i^{th} component of the diagonal embedding,

I_{Δ_{i}} = ⟨\overleftarrow{g_{i}}^{αβ} − \overrightarrow{g_{i}}^{αβ} | α, β = 1, …, r_{i}⟩,

simply sets the right and left pointing coordinates equal to each other in the i^{th} component. Choose a weight ω on the variables in the polynomial ring 𝕜[\overleftarrow{a_{i}}, \overrightarrow{a_{i}}] so that

ω(\overleftarrow{g_{i}}^{αβ}) = α − β and ω(\overrightarrow{g_{i}}^{αβ}) = β − α.

Using this weight we get a family

I_{Δ_{i}} = ⟨\overleftarrow{g_{i}}^{αβ} − \overrightarrow{g_{i}}^{αβ}t^{2α−2β} | α ≥ β⟩ + ⟨\overleftarrow{g_{i}}^{αβ}t^{2β−2α} − \overrightarrow{g_{i}}^{αβ} | α ≤ β⟩

of ideals over 𝔸^{1}, whose coordinate we call t.

Lemma 4.2. I_{Δ} = ∑_{i=0}^{n} I_{Δ_{i}} defines a flat group subscheme GL of GL^{2} × 𝔸^{1} whose special fiber at t = 0 is the group GL(0) = B_{+} ×_{T} B_{−} from (3.1).

Proof. The special fiber GL(0) over t = 0 is defined by I_{Δ}(0), which coincides with

in_{ω}(I_{Δ}) = ∑_{i=0}^{n}(⟨\overleftarrow{g_{i}}^{αβ} | α > β⟩ + ⟨\overrightarrow{g_{i}}^{αβ} | α < β⟩ + ⟨\overleftarrow{g_{i}}^{αα} − \overrightarrow{g_{i}}^{αα} | α = 1, …, r_{i}⟩)

by definition of the initial ideal in_{ω} for the weight ω [Eis95, Chapter 15]. Thus GL is a flat family (specifically, a Gröbner degeneration) over 𝔸^{1}. Flatness still holds after inverting the left and right pointing determinants.

It remains to show that GL is a subgroup scheme over 𝔸^{1}, i.e. that fiberwise multiplication in GL^{2} × 𝔸^{1} preserves GL ⊂ GL^{2} × 𝔸^{1}. This can be done set-theoretically because (for instance) all fibers of GL are geometrically reduced.

The fiber GL(0) over t = 0 is the subgroup B_{+} ×_{T} B_{−} ⊂ GL^{2}. On the other hand, the union of all fibers with t ≠ 0 is isomorphic to GL × (𝔸^{1} ∖ {0}) as a group subscheme of GL^{2} × (𝔸^{1} ∖ {0}), the isomorphism being

(4.1)  (\overleftarrow{γ}, \overrightarrow{γ}) × τ ↦ (τ.\overleftarrow{γ}τ.^{−1} , τ.^{−1}\overrightarrow{γ}τ.)

where τ. ∈ T has T (V_{i}) component τ_{i} with diagonal entries 1, τ, τ^{2}, …, τ^{r_{i}−1}. Indeed, the matrices (τ.\overleftarrow{γ}τ.^{−1} , τ.^{−1}\overrightarrow{γ}τ.) are precisely those satisfying the relations defining the ideal I_{Δ}, which can be rewritten more symmetrically as

I_{Δ}(t ≠ 0) = ⟨t^{−α} \overleftarrow{g_{i}}^{αβ} t^{β} − t^{α} \overrightarrow{g_{i}}^{αβ} t^{−β} | i = 0, …, n⟩

away from t = 0. □

4.2. Quiver degenerations

The orbits we wish to degenerate are of course the quiver loci. Recall the notation from Section 1.1 regarding the indexing on variables f^{i}_{αβ} in the coordinate ring 𝕜[Hom].

Definition 4.3. For p(f) ∈ 𝕜[Hom] and a coordinate t on 𝔸^{1} = 𝕜, let

p_{t}(f) = \frac{p(t^{α+β}f^{i}_{αβ})}{as much t as possible}.

Set I_{r} = ⟨p_{t}(f) | p(f) ∈ I_{r}⟩ ⊆ 𝕜[Hom × 𝔸^{1}], and let Ω_{r} ⊆ Hom × 𝔸^{1} be the zero scheme of I_{r}. For τ ∈ 𝕜 let Ω_{r}(τ) ⊂ Hom be the zero scheme of the ideal I_{r}(τ) ⊂ 𝕜[Hom] obtained from I_{r} by setting t = τ. The special fiber Ω_{r}(0) ⊂ Hom is called the quiver degeneration, and we refer to Ω_{r} as the family of Ω_{r}(0).

Lemma 4.4. The family Ω_{r} ⊆ Hom × 𝔸^{1} is flat over 𝔸^{1}.

Proof. The polynomial p_{t}(f) is obtained from p(f) by homogenizing with respect to the weight function on 𝕜[Hom] that assigns weight −α − β to the variable f^{i}_{αβ}. The resulting initial ideal is I_{r}(0), so [Eis95, Theorem 15.17] applies. □

Note that I_{r}(0) is rarely a monomial ideal. Thus Ω_{r}(1) ⇝ Ω_{r}(0) is only a “partial” Gröbner degeneration of Ω_{r} = Ω_{r}(1).

Proposition 4.5. The group scheme GL over 𝔸^{1} acts (fiberwise over 𝔸^{1}) on the flat family Ω_{r} over 𝔸^{1}.

Proof. This will follow from Proposition 4.1 applied to

𝔸^{n} = V, Γ_{0} = GL^{2}, \overline{A} = 𝔸^{1}, y = 0, \overline{G} = GL, and \overline{X} = Ω_{r}

by Lemmas 4.4 and 4.2, as soon as we show that GL(τ ≠ 0) acts on Ω_{r}(τ ≠ 0), these being the restrictions of GL and Ω_{r} to families over 𝔸^{1} ∖ {0}.

Resume the notation from the last paragraph of the proof of Lemma 4.2. Observe that Ω_{r} × (𝔸^{1} ∖ {0}) → Ω_{r}(τ ≠ 0) is an isomorphism of varieties under the map

(4.2)  (ϕ, τ) = ((ϕ_{i})_{i=1}^{n}, τ) ↦ (τ_{i−1}^{−1}ϕ_{i}τ_{i}^{−1}) =: τ.^{−1}ϕτ.^{−1}.

Now combine Eq. (4.1) and (4.2),

(τ.\overleftarrow{γ}τ.^{−1} , τ.^{−1}\overrightarrow{γ}τ.) · (τ.^{−1}ϕ τ.^{−1}) = (τ_{i−1}^{−1}\overrightarrow{γ}_{i−1}ϕ_{i}\overleftarrow{γ}_{i}^{−1}τ_{i}^{−1})
= τ.^{−1}[(\overleftarrow{γ}, \overrightarrow{γ}) · ϕ]τ.^{−1},

and use that (\overleftarrow{γ}, \overrightarrow{γ}) · ϕ ∈ Ω_{r} to get the desired result. □

Theorem 4.6, the main result of Section 4, relates the quiver degeneration Ω_{r}(0) to matrix Schubert varieties as in Proposition 3.4.

Theorem 4.6. Each irreducible component of Ω_{r}(0) is a possibly nonreduced product O(w) of matrix Schubert varieties for a lacing diagram w ∈ Hom.

Proof. By Proposition 4.5, the special fiber Ω_{r}(0) is stable under the action of GL(0) = B_{+} ×_{T} B_{−} on Hom, and hence is a union of orbit closures for this group. By Proposition 3.4 these orbit closures are direct products of matrix Schubert varieties. □

Remark 4.7. The methods of the current and the previous sections can be applied with no essential changes to the more general case of nilpotent cyclic quivers.

4.3. Multidegrees of quiver degenerations

Knowing multidegrees of quiver degenerations gives us multidegrees of quiver loci:

Lemma 4.8. The quiver degeneration Ω_{r}(0) has the same multidegree as Ω_{r}.

Proof. By Lemma 4.4 we may apply ‘Degeneration’ in [KM03a, Theorem 1.7.1]. □

Now we can conclude from Theorem 4.6 that the quiver polynomial has a positive expression in terms of products of double Schubert polynomials. This will provide a “lower bound” on the quiver polynomial Q_{r}(x − ^{∘}x), in Proposition 4.11.

Corollary 4.9. Let 𝔖_{w}(x − ^{∘}x) for a lacing diagram w denote the corresponding product ∏_{j=1}^{n} 𝔖_{w_{j}}(x^{j−1} − x^{j}) of double Schubert polynomials. Then

Q_{r}(x − ^{∘}x) = ∑_{w∈Hom} c_{w}(r) 𝔖_{w}(x − ^{∘}x),

where c_{w}(r) is the multiplicity of the component O(w) in Ω_{r}(0). In particular, c_{w}(r) ≥ 0.

Proof. In light of Lemma 4.8 and the additivity of multidegrees under taking unions of components of equal dimension [KM03a, Theorem 1.7.1], the quiver polynomial Q_{r}(x − ^{∘}x), which is by definition the multidegree of the quiver locus Ω_{r}, breaks up as a sum over lacing diagrams by Theorem 4.6. The summands are products of double Schubert polynomials by Theorem 2.1. □

To make Corollary 4.9 more explicit, it is necessary to identify which lacing diagrams w give components O(w), and to find their multiplicities. We shall accomplish this goal in Theorem 6.16, whose proof requires the less precise statement in Proposition 4.11, below, as a stepping stone (in the end, it turns out to have identified all the components, not just some of them). First, let us observe that the quiver degeneration Ω_{r}(0) contains every lacing diagram with rank array r.

Lemma 4.10. If the lacing diagram w has rank array r, then w ∈ Ω_{r}(τ) for all τ ∈𝕜.

Proof. Let w ∈ Hom have ranks r. It suffices to show that w ∈ Ω_{r}(τ) for τ ∈ 𝔸^{1} ∖ 0; indeed, it follows that it holds for all τ because the family Ω_{r} is closed in Hom × 𝔸^{1}.

By Lemma 3.2, w lies inside Ω_{r}. Using the notation of (4.1) and (4.2), for every τ ∈ 𝔸^{1}∖0 the list τ.wτ. of monomial matrices also lies in Ω_{r}. Now consider the curve

(τ.wτ.) × τ

inside Ω_{r} × (𝔸^{1} ∖ 0). Under the map given in (4.2), this curve is sent to the constant curve w × (𝔸^{1} ∖ 0) inside Ω_{r}(τ ≠ 0), proving the lemma. □

Proposition 4.11. Among the irreducible components of the quiver degeneration Ω_{r}(0) are the orbit closures O(w) for w ∈ W (r) (Definition 3.5). Hence c_{w}(r) ≥ 1 for w ∈ W (r).

Proof. Let w ∈ W (r). Then O(w) ⊂ Ω_{r}(0), by Lemma 4.10 along with Propositions 3.4 and 4.5. Consider a component of Ω_{r}(0) containing O(w); it must have the form O(w′) by Theorem 4.6. But O(w) ⊂ O(w′) implies that w′_{j} ≤ w_{j} in the strong Bruhat order for all j, by Lemma 3.3. Thus ℓ(w) ≥ ℓ(w′). But ℓ(w) is the codimension of O(w) in Hom for all w, and d(r) is the codimension of the flat degeneration Ω_{r}(0) of Ω_{r} in Hom. It follows that w′ = w. □

4.4. Rank stability of components

Later, in Section 6, it will be crucial to understand how the components of Ω_{r}(0) and their multiplicities behave when a constant m is added to all the ranks in r.

For notation, given m ∈ ℕ and a partial permutation w, let m + w be the new partial permutation obtained by letting w act on m + ℤ_{>0} = {m + 1, m + 2, …} instead of ℤ_{>0} = {1, 2, …} in the obvious manner. In particular m + w fixes 1, 2, …, m. For a list w = (w_{1}, …, w_{n}) of partial permutations, set m+w = (m+w_{1}, …, m+w_{n}). If m is a nonnegative integer and r = (r_{ij}) is a rank array, let m + r be the new rank array obtained by adding m to every rank r_{ij} in the array r. This transformation is accomplished simply by adding m to the bottom entry s_{0n} in the lace array s appearing in (1.2).

In dealing with stability of components of quiver degenerations Ω_{r}(0), we take our cue from the set W (r), whose behavior in response to uniformly increasing ranks we deduce immediately from Theorem 3.8.

Corollary 4.12. There is a bijection W (r) → W (m + r) given by adding m (0, n)-laces along the bottom.

The corresponding statement for the components of Ω_{r}(0) is also true; moreover, the multiplicities of these components are also stable. In fact, the entire geometry of the family Ω_{m+r} is essentially the same as the geometry of the family Ω_{r}.

Proposition 4.13. The orbit closure O(w) is a component of the quiver degeneration Ω_{r}(0) if and only if O(m + w) is a component of the quiver degeneration Ω_{m+r}(0), and the multiplicities at their generic points are equal: c_{w}(r) = c_{m+w}(m + r). Furthermore, every component of Ω_{m+r}(0) has the form O(m + w) for some w ∈ Ω_{r}(0).

Proof. By induction on m, it suffices to prove the case m = 1 (the case m = 0 being trivial).

Fix a rank array r. Let Hom′ denote the vector space of matrix lists ψ = (ψ_{1}, …, ψ_{n}) in which ψ_{i} has size (1 + r_{i−1}) × (1 + r_{i}) for i = 1, …, n. Just as GL acts on Hom, let GL′ act on Hom′ via the diagonal embedding Δ : GL′ ↪ (GL′)^{2}.

Next we define two subgroups R′ and C′ of GL′. The “row” group R′ consists of matrix lists ρ′ = (ρ′_{0}, …, ρ′_{n}) ∈ GL′ such that ρ′_{0} is the identity matrix, while ρ′_{i} for i ≥ 1 has 1’s on its diagonal and is otherwise nonzero only in its bottom row. Similarly, the “column” group C′ consists of matrix lists κ′ = (κ′_{0}, …, κ′_{n}) ∈ GL′ such that κ′_{n} is the identity matrix, while κ′_{i} for i ≤ n − 1 has 1’s on its diagonal and is otherwise nonzero only in its rightmost column. By conjugation as in (4.1), we get two corresponding group schemes C′ and R′ over the affine line 𝔸 = 𝔸^{1}(𝕜), just as we get GL′ from GL′.

Let Hom, 𝕜^{∗}, and Hom′ denote the products of the corresponding (un-tilded) spaces with the line 𝔸, where 𝕜^{∗} = 𝔸^{1} ∖ {0} is the multiplicative group of 𝕜. In addition, for λ = (λ_{1}, …, λ_{n}) ∈ (𝕜^{∗})^{n}, write ϕ ⊕ λ ∈ Hom′ for the matrix list whose i^{th} component has ϕ_{i} in the northwest corner and λ_{i} in the southeast corner. Now define the morphism

θ : C′ ×_{𝔸} R′ ×_{𝔸} Hom ×_{𝔸} (𝕜^{∗})^{n} ⟶ Hom′

by letting θ take the fiber over τ ∈ 𝔸 to its image via

θ : (κ′, ρ′, ϕ, λ) ⟼ κ′ · ρ′ · (ϕ ⊕ λ).

The dots after κ′ and ρ′ denote the action of GL′ on Hom′.

Our first task is to show that θ is an isomorphism onto its (open) image. This is completely elementary, but since it will be useful to know the image of θ precisely, let us say how to construct the inverse. Some notation will help. For ρ′ ∈ R′(τ), write

ρ′_{i} = (1 + \overleftarrow{ρ}_{i}, 1 + \overrightarrow{ρ}_{i}),

where 1 is the identity matrix of size 1 + r_{i}. Thus, for example, \overleftarrow{ρ}_{i} has at most one nonzero row, namely its bottom row, and 1 + \overleftarrow{ρ}_{i} acts via its inverse, which equals 1 − \overleftarrow{ρ}_{i}. Similarly, for κ′ ∈ C′(τ), write κ′_{i} = (1 + \overleftarrow{κ}_{i}, 1 + \overrightarrow{κ}_{i}). We identify λ ∈ (𝕜^{∗})^{n} with the matrix list 0 ⊕ λ ∈ Hom′, writing |λ_{i}| if we wish to refer to the sole nonzero entry of λ_{i}. Finally, we identify Hom with the subvariety of Hom′ consisting of all matrix lists ψ such that each ψ_{i} is zero in its rightmost column as well as its bottom row; thus ϕ ∈ Hom is identified with “ϕ ⊕ 0” (we only defined ϕ ⊕ λ for nonzero λ). In this notation,

(4.3) θ(κ′, ρ′, ϕ, λ) = ψ, where ψ_{i} = [ \begin{matrix}ϕ_{i} + \overrightarrow{κ}_{i−1}v_{i} & \overrightarrow{κ}_{i−1}λ_{i} − (ϕ_{i} + \overrightarrow{κ}_{i−1}v_{i})\overleftarrow{κ}_{i} \\ v_{i} & λ_{i} − v_{i}\overleftarrow{κ}_{i} \end{matrix} ]
and v_{i} = \overrightarrow{ρ}_{i−1}ϕ_{i} − λ_{i}\overleftarrow{ρ}_{i}.

As in the above display, each matrix ψ_{i} in a matrix list ψ ∈ Hom′ can be expressed as ψ_{i} = NW (ψ_{i}) + NE(ψ_{i}) + SE(ψ_{i}) + SW (ψ_{i}), where

NW (ψ_{i}) = [ \begin{matrix}∗ & 0 \\ 0 & 0 \end{matrix} ] , NE(ψ_{i}) = [ \begin{matrix}0 & ∗ \\ 0 & 0 \end{matrix} ] , SE(ψ_{i}) = [ \begin{matrix}0 & 0 \\ 0 & ∗ \end{matrix} ] , and SW (ψ_{i}) = [ \begin{matrix}0 & 0 \\ ∗ & 0 \end{matrix} ].

The ∗ blocks agree with ψ_{i} in rectangles of size (r_{i−1} × r_{i}), (r_{i−1} × 1), (1 × 1), and (1 × r_{i}), in the order listed above. The morphism θ is inverted by an easy seven-step program. For the first five steps, proceed from i = n down to i = 1; each loop here assumes that \overleftarrow{κ}_{i} has been defined in the previous loop, starting from \overleftarrow{κ}_{n} = 0.

1. Set v_{i} = SW (ψ_{i}).

2. Set λ_{i} = SE(ψ_{i}) + v_{i}\overleftarrow{κ}_{i}.

3. Set \overrightarrow{κ}_{i−1} = \frac{1}{|λ_{i}|}(NE(ψ_{i}) + NW (ψ_{i})\overleftarrow{κ}_{i}).

4. Set \overleftarrow{κ}_{i−1} = τ_{i−1}^{2}\overrightarrow{κ}_{i−1}τ_{i−1}^{−2} if τ ≠ 0 and \overleftarrow{κ}_{i−1} = 0 if τ = 0 (τ_{i} is defined in (4.1)).

5. Set ϕ_{i} = NW (ψ_{i}) − \overrightarrow{κ}_{i−1}v_{i}. To complete the inversion, proceed in a loop from i = 1 up to i = n, starting with \overrightarrow{ρ}_{0} = 0:

6. Set \overleftarrow{ρ}_{i} = −\frac{1}{|λ_{i}|}(v_{i} − \overrightarrow{ρ}_{i−1}ϕ_{i}).

7. Set \overrightarrow{ρ}_{i} = τ_{i}^{−2}\overleftarrow{ρ}_{i}τ_{i}^{2} if τ ≠ 0 and \overrightarrow{ρ}_{i} = 0 if τ = 0.

The conjugation in Step 7 multiplies each entry of \overleftarrow{ρ}_{i} by a strictly positive power of τ, so the definition of \overrightarrow{ρ}_{i} is polynomial in τ, even near τ = 0; the same goes for \overleftarrow{κ}_{i−1}. Step 2 defines the λ coordinates as rational functions on Hom′. In fact, λ_{i}λ_{i+1} ⋯ λ_{n} is a polynomial function on Hom′ for each i = 1, …, n. The algorithm produces the inverse of the morphism θ on the locus in Hom′ where these polynomials are nonzero, regardless of which fiber τ is being considered. The elementary details of this verification are omitted.

Let θ_{r} be the restriction of θ to the subvariety where the Hom component is replaced by Ω_{r}. Our next task is to prove that θ_{r} has image Ω_{1+r}^{∘}, by which we mean the intersection of Ω_{1+r} with the image of θ, where λ_{1} ⋯ λ_{n} is nonzero. As θ is an isomorphism, the image of θ_{r} is Zariski closed in the image of θ and reduced, since its source is. Moreover, the dimension of the source of θ_{r} is dim(Ω_{r}) + dim(R′) + dim(C′) + 1 = dim(Ω_{1+r}). Hence it is enough to show that the image of θ_{r} is contained in Ω_{1+r}. Again by Zariski closedness we need only check this over fibers τ ≠ 0. Finally, since θ was constructed to be equivariant for the conjugation action defining Ω_{r}, R′, C′, and Ω_{1+r}, we need only check the fiber τ = 1. This final step is elementary: if ϕ ∈ Ω_{r}, then ϕ ⊕ λ ∈ Ω_{1+r} by definition; our task is done because θ_{r} spreads ϕ ⊕ λ around inside of Hom′ via the action of GL′, which preserves Ω_{1+r}.

Our third task is to identify the components (and their multiplicities) in the zero fiber of the image of θ_{r}, which we have shown is Ω_{1+r}^{∘}(0). The isomorphism of C′×_{𝔸}R′×_{𝔸}Ω_{r}×_{𝔸}(𝕜^{∗})^{n} with Ω_{1+r}^{∘} immediately implies that the components of Ω_{1+r}^{∘}(0) are in bijection with those of Ω_{r}(0), and that the corresponding multiplicities are equal. To complete this task, it therefore remains only to be more explicit about the bijection.

Let θ_{r}(0) be the restriction of θ_{r} to the fiber over τ = 0. Referring to (4.3), the fact that \overleftarrow{κ}_{i} is zero over τ = 0 immensely simplifies θ_{r}(0). Writing [v_{i}, λ_{i}] for the bottom row of ψ_{i} and [{κ_{i−1} \atop 1} ] for the single column κ_{i−1} with an extra 1 at the bottom, the effect of θ_{r}(0) is to add the rank 1 matrix [v_{i}, λ_{i}] · [{κ_{i−1} \atop 1} ] to ϕ_{i}.

Let O(w′) be the component of Ω_{1+r}(0) corresponding via θ_{r}(0) to a component O(w) of Ω_{r}(0); thus O(w′) is Zariski closed in all of Hom′, not just where λ_{1} ⋯ λ_{n} ≠ 0. Since v_{i}, λ_{i}, and κ_{i−1} can be chosen arbitrarily as long as λ_{i} ≠ 0, the matrix Schubert variety \overline{X}_{w_{i}′} (Definition 1.11) appearing as a factor of O(w′) contains the sum of the r_{i−1} × r_{i} matrix Schubert variety \overline{X}_{w_{i}} with the variety of all (1 + r_{i−1}) × (1 + r_{i}) matrices of rank at most 1. But rank(Z_{q×p}) ≤ rank((1 + w_{i})_{q×p}) for all matrices Z in this sum, and equality is attained whenever a sufficiently generic matrix of rank 1 is added to a full-rank matrix in \overline{X}_{w_{i}}. It follows that O(w′) contains O(1 + w), so O(w′) = O(1 + w) by dimension considerations.

Our final task is to verify that no components of Ω_{1+r}(0) have been lost by restricting to Ω_{1+r}^{∘} instead of Ω_{1+r}. More precisely, we need to prove that no component of Ω_{1+r}(0) lies in the zero set of the polynomial λ_{1} ⋯ λ_{n}. Our task is made easier by the fact that the zero set of λ_{1} ⋯ λ_{n} over τ = 0 agrees with that of the product f = ∏_{i=1}^{n} f^{i}_{1+r_{i−1}, 1+r_{i}} of the southeast entries of the matrices f^{i} of coordinate variables (Definition 1.1); this fact is immediate from Step 2, above, which implies that λ_{i} = f^{i}_{1+r_{i−1}, 1+r_{i}} over τ = 0.

The key point in checking the nonzerodivisor property is the following general statement.

Lemma 4.14. Suppose that I ⊆ 𝕜[f^{1}, …, f^{n}] is a prime ideal generated by polynomials homogeneous in each of the finite alphabets f^{1}, …, f^{n}. Assume that I contains none of the ideals ⟨f^{1}⟩, …, ⟨f^{n}⟩ generated by a complete alphabet. If P is a minimal prime of some initial monomial ideal in(I) of I, then P contains none of the ideals ⟨f^{1}⟩, …, ⟨f^{n}⟩.

Proof. Let 𝔸 = 𝔸_{1} × ⋯ × 𝔸_{n}, where 𝔸_{j} = Spec(𝕜[f^{j}]). Using the homogeneity hypothesis, the zero scheme Ω ⊆ 𝔸 of I has a multidegree C(Ω) ∈ ℤ[x_{1}, …, x_{n}], where x_{i} corresponds to ordinary total degree in the variables f^{i}. The zero set of each minimal prime of in(I) is a linear subspace L ⊆ 𝔸 of the same dimension as Ω itself [KS95], with multidegree C(L) and multiplicity μ(L) in the zero scheme of in(I). By additivity [KM03a, Theorem 1.7.1], C(Ω) is a sum of terms μ(L) · C(L), one for every such L.

On the other hand, the hypotheses on I imply that it defines a closed subvariety of the corresponding product ℙ = ℙ_{1} × ⋯ × ℙ_{n} of projective spaces. Viewing the Gröbner degeneration as taking place inside of ℙ, an irreducible component L ⊆ 𝔸 of the (reduced) zero set of in(I) contributes a component to the special fiber if and only if its defining ideal contains none of the ideals ⟨f^{1}⟩, …, ⟨f^{n}⟩. Since the multidegree C(Ω) equals the subsum of the terms μ(L) · C(L) coming from the components L contributing to the projective special fiber, we conclude that every component L must contribute. □

The special fiber of any Gröbner degeneration of Ω_{1+r}(0) can be realized as the special fiber of a Gröbner degeneration of Ω_{1+r} itself. Therefore, Gröbner degenerating Ω_{1+r}(0) to the zero scheme Ω′ of a monomial ideal, we find by Lemma 4.14 applied to I = I_{1+r} that no component of Ω′—and hence no component O(w′) of Ω_{1+r}(0)—is defined by an ideal containing ⟨f^{i}⟩. Therefore the partial permutation w′_{i} is nonzero for all i = 1, …, n if O(w′) is a component of Ω_{1+r}(0). The condition w′_{i} ≠ 0 means that \overline{X}_{w_{i}′} has dimension ≥ 1. The southeast corner variable is a nonzerodivisor on every positive-dimensional matrix Schubert variety. Hence the product f is a nonzerodivisor on every component of Ω_{1+r}(0). By the argument preceding Lemma 4.14, we conclude that the same holds for λ_{1} ⋯ λ_{n}. □

Remark 4.15. It would be possible at this juncture to justify replacing the double Schubert polynomials in Corollary 4.9 with double Stanley symmetric functions, since our results through Proposition 4.13 suffice for this purpose. (Notably, the ratio formula does not enter into these arguments, which appear in the proofs of Theorem 6.16 and Corollary 6.23, below.) Hence we could immediately deduce the positivity of the Buch–Fulton quiver constants c_{\underline{λ}}(r) using the Schur-positivity of Stanley symmetric functions. However, we postpone this discussion until after we have reviewed Stanley symmetric functions and their Schur-positivity in Section 6, because we shall in any case prove more precise statements about the quiver constants in Sections 7 and 8.

Section 5. Pipe dreams for Zelevinsky permutations

5.1. Pipe dream formula for double quiver polynomials

Schubert polynomials possess a beautiful combinatorial description, proved independently in [BJS93, FS94], in terms of diagrams defined in [FK96] that we call ‘reduced pipe dreams’, although they have also been called ‘rc-graphs’ in the literature [BB93, Len02]. These diagrams are the main characters in our next combinatorial formula, Theorem 5.5, which is little more than a corollary of the ratio formula. Although it is also of independent interest, Theorem 5.5 is an essential ingredient for proving our remaining formulae. From a purely combinatorial perspective, we find the direct connection between reduced pipe dreams for Zelevinsky permutations and lacing diagrams particularly appealing (Section 5.2).

Consider a square grid of size ℓ × ℓ, with the box in row q and column p labeled (q, p), as in ℓ × ℓ matrix. If each box in the grid is covered with a square tile containing either or [figure] , then one can think of the tiled grid as a network of pipes.

Definition 5.1. A pipe dream is a subset of the ℓ × ℓ grid, identified as the set of crosses in a tiling by crosses and elbow joints [figure] . A pipe dream is reduced if each pair of pipes crosses at most once. The set RP(v) of reduced pipe dreams for a permutation v ∈ S_{ℓ} is the set of reduced pipe dreams D such that the pipe entering row q flows northeast to exit from column v(q).

We usually draw crossing tiles as some sort of cross, either ‘+’ or ‘ ’, although we shall frequently use an asterisk ∗ to denote a strictly above the block superantidiagonal. We often leave elbow tiles blank or denote them by dots, to make the diagrams less cluttered. We shall only be interested in pipe dreams whose entries lie in the triangular region strictly above the main antidiagonal (in spots (q, p) with q + p ≤ ℓ), with elbow joints elsewhere in the square grid of size ℓ.

Example 5.2. Using ℓ = 12, a typical reduced pipe dream for the Zelevinsky permutation v in Example 1.9 (v is written down the left side) looks like

\begin{matrix}8 & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & · & · & + & · & · \\ 9 & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & · & · & · & · & · \\ 4 & ∗ & ∗ & ∗ & · & · & + & + & · & · & · & · & · \\ 5 & ∗ & ∗ & ∗ & · & · & · & · & + & · & · & · & · \\ 11 & ∗ & ∗ & ∗ & · & · & · & · & · & · & · & · & · \\ 1 & · & · & + & · & + & · & · & · & · & · & · & · \\ 2 & · & · & · & · & · & · & · & · & · & · & · & · \\ 6 & · & + & · & · & · & · & · & · & · & · & · & · \\ 12 & · & · & · & · & · & · & · & · & · & · & · & · \\ 3 & · & · & · & · & · & · & · & · & · & · & · & · \\ 7 & · & · & · & · & · & · & · & · & · & · & · & · \\ 10 & · & · & · & · & · & · & · & · & · & · & · & · \end{matrix} ⟷ [figure: 8; ∗; ∗; ∗; ∗; ∗; ∗; ∗; 9; ∗; ∗; ∗; ∗; ∗; ∗; ∗; 4; ∗; ∗; ∗; 5; ∗; ∗; ∗; 11; ∗; ∗; ∗; 1; 2; 6; 12; 3; 7; 10]

when the ∗’s remain as in the diagram. Although each ∗ represents a in every pipe dream for v, the ∗’s will be just as irrelevant here as they were for the diagram. □

Each pipe dream D yields a monomial (X − Y)^{D}, defined as the product over all entries in D of (x_{+} − y_{+}), where x_{+} sits at the left end of the row containing , and y_{+} sits atop the column containing . At this level of generality, X and Y are simply alphabets (of size ℓ or of infinite size; it does not matter). Here is the fundamental result relating pipe dreams to Schubert polynomials; its ‘single’ form was proved independently in [BJS93, FS94], but it is the ‘double’ form that we need.

Theorem 5.3. [FK96] The double Schubert polynomial for v equals the sum

𝔖_{v}(X − Y) = ∑_{D∈RP(v)}(X − Y)^{D}

of all monomials associated to reduced pipe dreams for v.

In pictures of pipe dreams D for Zelevinsky permutations, we retain the notation from Section 2 regarding block matrices of total size d. For the purposes of seeing where crosses lie, we will not need the ‘∗’ entries above the block superantidiagonal of D. In other words, we shall consider the crosses in the pipe dream D ∖ D_{Hom} instead of D, where D_{Hom} is identified with the reduced pipe dream consisting entirely of ∗ locations (the unique reduced pipe dream for the dominant Zelevinsky permutation of Hom).

Example 5.4. The pipe dream from Example 5.2 has row and column labels

\begin{aligned}& y_{1}^{3} & y_{2}^{3} & y_{3}^{3} & y_{1}^{2} & y_{2}^{2} & y_{3}^{2} & y_{4}^{2} & y_{1}^{1} & y_{2}^{1} & y_{3}^{1} & y_{1}^{0} & y_{2}^{0} \\ x_{1}^{0} & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & · & · & + & · & · \\ x_{2}^{0} & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & · & · & · & · & · \\ x_{1}^{1} & ∗ & ∗ & ∗ & · & · & + & + & · & · & · & · & · \\ x_{2}^{1} & ∗ & ∗ & ∗ & · & · & · & · & + & · & · & · & · \\ x_{3}^{1} & ∗ & ∗ & ∗ & · & · & · & · & · & · & · & · & · \\ x_{1}^{2} & · & · & + & · & + & · & · & · & · & · & · & · \\ x_{2}^{2} & · & · & · & · & · & · & · & · & · & · & · & · \\ x_{3}^{2} & · & + & · & · & · & · & · & · & · & · & · & · \\ x_{4}^{2} & · & · & · & · & · & · & · & · & · & · & · & · \\ x_{1}^{3} & · & · & · & · & · & · & · & · & · & · & · & · \\ x_{2}^{3} & · & · & · & · & · & · & · & · & · & · & · & · \\ x_{3}^{3} & · & · & · & · & · & · & · & · & · & · & · & · \end{aligned} = \begin{aligned}& ḋ_{1} & ḋ_{2} & ḋ_{3} & ċ_{1} & ċ_{2} & ċ_{3} & ċ_{4} & ḃ_{1} & ḃ_{2} & ḃ_{3} & ȧ_{1} & ȧ_{2} \\ a_{1} & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & · & · & + & · & · \\ a_{2} & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & · & · & · & · & · \\ b_{1} & ∗ & ∗ & ∗ & · & · & + & + & · & · & · & · & · \\ b_{2} & ∗ & ∗ & ∗ & · & · & · & · & + & · & · & · & · \\ b_{3} & ∗ & ∗ & ∗ & · & · & · & · & · & · & · & · & · \\ c_{1} & · & · & + & · & + & · & · & · & · & · & · & · \\ c_{2} & · & · & · & · & · & · & · & · & · & · & · & · \\ c_{3} & · & + & · & · & · & · & · & · & · & · & · & · \\ c_{4} & · & · & · & · & · & · & · & · & · & · & · & · \\ d_{1} & · & · & · & · & · & · & · & · & · & · & · & · \\ d_{2} & · & · & · & · & · & · & · & · & · & · & · & · \\ d_{3} & · & · & · & · & · & · & · & · & · & · & · & · \end{aligned}

where we use alphabets a, b, c, d interchangeably with x^{0}, x^{1}, x^{2}, x^{3}, and also ḋ, ċ, ḃ, ȧ interchangeably with y^{3}, y^{2}, y^{1}, y^{0}. The monomial for the above pipe dream is

(a_{1} − ḃ_{3})(b_{1} − ċ_{3})(b_{1} − ċ_{4})(b_{2} − ḃ_{1})(c_{1} − ḋ_{3})(c_{1} − ċ_{2})(c_{3} − ḋ_{2}),

ignoring all ∗ entries as required. Removing the dots yields this pipe dream’s contribution to the ordinary quiver polynomial. □

Now we have enough ingredients for our second positive combinatorial formula for double quiver polynomials, the first being Definition 2.5.

Theorem 5.5. The double quiver polynomial for ranks r equals the sum

Q_{r}(x − ^{∘}y) = ∑_{D∈RP(v(r))}(x_{r} − ^{∘}y_{r})^{D∖D_{Hom}}

over all reduced pipe dreams D for the Zelevinsky permutation v(r) of the monomials (x_{r} − ^{∘}y_{r})^{D∖D_{Hom}} made from crosses occupying the block antidiagonal and block superantidiagonal (that is, ignoring ∗ entries occupying boxes strictly above the block superantidiagonal).

Proof. This follows from Definition 2.5 and Theorem 5.3, using the fact that every pipe dream D ∈ RP(v(r)) contains the subdiagram D_{Hom}, and that RP(v(Hom)) consists of the single pipe dream D_{Hom}. □

In other words, the double quiver polynomial Q_{r}(x − ^{∘}y) is the sum over monomials for all “skew reduced pipe dreams” D ∖ D_{Hom} with D ∈ RP(v(r)).

Since it will be important in Section 5.3 (and adds even more concreteness to Theorem 5.3), let us provide one way of generating all reduced pipe dreams for v.

Definition 5.6 ([BB93]). A chutable rectangle is a connected 2 × k rectangle C inside a pipe dream D such that k ≥ 2 and every location in C is a except the following three: the northwest, southwest, and southeast corners. Applying a chute move to D is accomplished by placing a in the southwest corner of a chutable rectangle C and removing the from the northeast corner of the same C.

Heuristically, a chute move therefore looks like:

\begin{aligned}& & & & & & & & ... \\ & · & + & + & … & + & + & + \\ & · & + & + & & + & + & · \\ ... \end{aligned} ⇝^{chute}  \begin{aligned}& & & & & & & & ... \\ & · & + & + & … & + & + & · \\ & + & + & + & & + & + & · \\ ... \end{aligned}

Proposition 5.7. [BB93] There is a unique top reduced pipe dream D_{top} for v, in which no has any elbows [figure] due north of it. Every reduced pipe dream in RP(v) can be obtained by starting with D_{top} and applying some sequence of chute moves.

5.2. Pipes to laces

Pipe dreams for Zelevinsky permutations give rise to lacing diagrams.

Definition 5.8. The j^{th} antidiagonal block is the block of size r_{j} × r_{j} along the main antidiagonal in the j^{th} block row. Given a reduced pipe dream D for the Zelevinsky permutation v(r), define the partial permutation w_{j} = w_{j}(D) sending p to q if the pipe entering the p^{th} column from the right of the (j − 1)^{st} antidiagonal block enters the j^{th} antidiagonal block in its q^{th} column from the right. Set w(D) = (w_{1}, …, w_{n}). Equivalently, w(D) is the lacing diagram determined by D.

Example 5.9. The partial permutations arising from the reduced pipe dream in Examples 5.2 and 5.4 come from the following partial reduced pipe dreams,

[figure: 2 1; 8; 9; 3 2 1; 3 2 1; 4; 5; 11; 4 3 2 1; 4 3 2 1; 1; 2; 6; 12; 3 2 1] (5.1)

where they send each number along the top to the number along the bottom connected to it by a pipe (if such a pipe exists), or to nowhere. It is easy to see the lacing diagram from these pictures. Indeed, removing all segments of all pipes not contributing to one of the partial permutations leaves some pipes

[figure: 8; 9; 4; 5; 11; 1; 2; 6; 12; 3; 7; 10]  ≈  a b c d

that can be interpreted directly as the desired lacing diagram

a b c d

by shearing to make the rightmost dots in each row line up vertically, and then reflecting through the diagonal line ⧹ of slope −1. □

Warning: the d × d ‘pipe’ representation of the lacing diagram (at left above) does not determine a reduced pipe dream uniquely, because although the ordinary and singly virtual crossings have been placed in the d × d grid, the doubly virtual crossings still need to be placed (in the lower two superantidiagonal blocks).

Theorem 5.10. Every reduced pipe dream D ∈ RP(v(r)) gives rise to a minimally crossing lacing diagram w(D) with rank array r; that is, w(D) ∈ W (r).

Proof. Each × entry in the permutation matrix for v(r) corresponds to a pipe in D entering due north of it and exiting due west of it. The permutation v(r) was specifically constructed to have exactly s_{ij} entries × (for i, j = 0, …, n) in the intersection of the i^{th} block row and the j^{th} block column from the right. □

We shall see in Corollary 6.18 that in fact every minimally crossing lacing diagram arises in this manner from a reduced pipe dream.

5.3. Rank stability of pipe dreams

Just as components in the quiver degeneration are stable for uniformly increasing ranks r (Proposition 4.13), so are the sets of pipe dreams for v(r). Before stating the precise result, we analyze the changes to v(r) and its diagram when 1 is added to all ranks in r.

Lemma 5.11. The passage from r to 1 + r enlarges every block submatrix in the permutation matrix for v(r) by one row and one column. Similarly, the diagram of v(1 + r) has the same rectangular collections of cells in the southeast corners of the same blocks as does the diagram of v(r). (For an example, see Fig. 2.)

\begin{matrix}10 & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & × & · & · & · & · & · & · \\ 11 & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & · & × & · & · & · & · & · \\ 12 & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & · & · & × & · & · & · & · \\ 5 & ∗ & ∗ & ∗ & ∗ & × & · & · & · & · & · & · & · & · & · & · & · \\ 6 & ∗ & ∗ & ∗ & ∗ & · & × & · & · & · & · & · & · & · & · & · & · \\ 7 & ∗ & ∗ & ∗ & ∗ & · & · & × & · & · & · & · & · & · & · & · & · \\ 14 & ∗ & ∗ & ∗ & ∗ & · & · & · & □ & □ & · & · & · & □ & × & · & · \\ 1 & × & · & · & · & · & · & · & · & · & · & · & · & · & · & · & · \\ 2 & · & × & · & · & · & · & · & · & · & · & · & · & · & · & · & · \\ 3 & · & · & × & · & · & · & · & · & · & · & · & · & · & · & · & · \\ 8 & · & · & · & □ & · & · & · & × & · & · & · & · & · & · & · & · \\ 15 & · & · & · & □ & · & · & · & · & □ & · & · & · & □ & · & × & · \\ 4 & · & · & · & × & · & · & · & · & · & · & · & · & · & · & · & · \\ 9 & · & · & · & · & · & · & · & · & × & · & · & · & · & · & · & · \\ 13 & · & · & · & · & · & · & · & · & · & · & · & · & × & · & · & · \\ 16 & · & · & · & · & · & · & · & · & · & · & · & · & · & · & · & × \end{matrix}

Figure 2. The permutation v(1 + r) and its diagram, for v(r) as in Example 1.9

Proof. The lace array giving rise to 1 + r is the same as the array s giving rise to r except that s_{0n} must be replaced by 1 + s_{0n}. The rest is straightforward. □

Suppose D ∈ RP(v(r)), and split D into horizontal strips (block rows). In each strip, chute every upward (that is, apply inverse chute moves) as far as possible. By Proposition 5.7, this yields a top reduced pipe dream in each horizontal strip. Since every relevant in a pipe dream D ∈ RP(v(r)) is confined to the antidiagonal blocks and those immediately to their left, we consider the j^{th} horizontal strip of a pipe dream to consist only of those two blocks, with a single vertical dividing line.

Example 5.12. For example, in our running example of a reduced pipe dream (Examples 5.2, 5.4, and 5.9), the top pipe dream in the second (that is, j = 2) horizontal strip is depicted on the left side of Fig. 3. The corresponding top pipe dream (see Proposition 5.15)

\begin{aligned}& ḋ_{1} & ḋ_{2} & ḋ_{3} & ċ_{1} & ċ_{2} & ċ_{3} & ċ_{4} \\ c_{1} & · & · & + & · & + & · & · \\ c_{2} & · & · & + & · & · & · & · \\ c_{3} & · & · & · & · & · & · & · \\ c_{4} & · & · & · & · & · & · & · \end{aligned}  \begin{aligned}& ḋ_{1} & ḋ_{2} & ḋ_{3} & ḋ_{4} & ċ_{1} & ċ_{2} & ċ_{3} & ċ_{4} & ċ_{5} \\ c_{1} & · & · & · & + & · & + & · & · & · \\ c_{2} & · & · & · & + & · & · & · & · & · \\ c_{3} & · & · & · & · & · & · & · & · & · \\ c_{4} & · & · & · & · & · & · & · & · & · \\ c_{5} & · & · & · & · & · & · & · & · & · \end{aligned}

Figure 3. Top pipe dreams in D ∈ RP(v), for v = v(r) or v(1 + r)

in that horizontal strip for v(1 + r) is on the right. □

Lemma 5.13. Suppose D ∈ RP(v(r)) consists of miniature top reduced pipe dreams in its horizontal strips. Inside the restriction of D to its superantidiagonal block in block row j, the 2 × 2 configurations \begin{matrix}+ & + \\ + & · \end{matrix} and \begin{matrix}∗ & ∗ \\ + & · \end{matrix} in consecutive rows and columns never occur.

Proof. Suppose there is such a 2 × 2 configuration. Since the restriction of D to the superantidiagonal block in block row j is a top pipe dream, all the cells in D due north of the 2 × 2 configuration are either or ∗, and all of these represent crossings in D. Therefore the southwest in the 2 × 2 configuration represents the crossing of two pipes exiting through adjacent columns in block column j + 1 from the right. This is impossible because the transpose of D is a reduced pipe dream for the permutation v(r)^{−1}, which is the Zelevinsky permutation v(r′) for the transpose ranks r′ and hence can have no descents within a block row. □

The easy proof of the next lemma is omitted.

Lemma 5.14. Adding a row of elbows across the top (or a column of elbows to the left) of any reduced pipe dream for a permutation w yields a reduced pipe dream for 1 + w. □

Here is the rank stability statement for pipe dreams, to be applied in Section 6.

Proposition 5.15. Let top(r) ⊆ RP(v(r)) denote the set of reduced pipe dreams D for v(r) consisting of a miniature top pipe dream D_{j} in the j^{th} horizontal strip, for j = 0, …, n − 1. There is a canonical bijection top(r) ≅ top(m + r) for all m ≥ 0. Moreover, corresponding miniature top pipe dreams D_{j} and D′_{j} are translates of each other, in the same relative position to the vertical dividing line in horizontal strip j.

Proof. It suffices by induction to prove the case m = 1. Starting with D ∈ top(r), inflate the blocks as in Lemma 5.11, and construct a new pipe dream D′ by placing each miniature top pipe dream D_{j} in the same location relative to the vertical dividing line in its inflated strip. First we claim that D′ ∈ RP(v(1 + r)), for then automatically D′ ∈ top(1 + r). Then we claim that every reduced pipe dream in top(1 + r) comes from some D ∈ top(r) in this manner.

What we show precisely for the first claim is that for every × in the permutation matrix for v(1 + r), the pipe (in D′) entering into its row exits through its column. Given a × entry of v(1 + r), exactly one of the following three statements must hold, by construction of Zelevinsky permutations: (i) the × lies in the northwest corner of some superantidiagonal block; (ii) the × lies in the southeast corner of the whole matrix; or (iii) there is a corresponding × in v(r).

For an × entry of type (i), the pipe entering its row encounters precisely one elbow, located in the same position as ×, that turns the pipe north to exit through its column. For the × of type (ii), observe that every entry along the superantidiagonal (not the block superantidiagonal) is an elbow, because of the extra row added into each block row (Lemma 5.11). Thus its pipe proceeds from the southwest corner to the northeast with no obstructions.

For an × entry of type (iii), we claim that the pipe P′ starting in the row of × enters the first antidiagonal block (through which it passes) in the same location relative to the block’s northwest corner as does the pipe P for the corresponding × in v(r). If the × lies in the bottom block row, then this is trivial, and if P′ never enters an antidiagonal block, then this is vacuous. Otherwise, the × itself lies one row farther south from the top of its block row, and we must show why this difference is recovered by P′. Think of each D′_{j} as having a blank column (no entries) along the western edge of its strip, and a blank row along the southern edge of its strip. The pipe P′ recovers one unit of height as it passes through each such region, and it passes through exactly one on the way to its first antidiagonal block.

Suppose that both P and P′ enter an antidiagonal block at the same vertical distance from the block’s northwest corner. Since D′ agrees with D in each antidiagonal block (as measured from their northwest corners), the pipe P′ exits the antidiagonal block in the same position relative to its entry point as does P. Therefore we can use induction on the number of antidiagonal blocks traversed by P to show that the pipe P′ exits every antidiagonal block in the same location relative to its northwest corner as does P. If the × lies in the rightmost block column, then P′ has already finished in the correct column. Otherwise, P′ must recover one more horizontal unit, which it does by traversing the blank row and column in the top horizontal strip.

For the second claim, we need only show that for any given D′ ∈ top(1 + r), each horizontal strip in D′ has a column of elbows along its western edge, a row of elbows along its southern edge, and an antidiagonal’s worth of elbows on the superantidiagonal, for then each miniature top pipe dream D′_{j} fits inside the deflated block matrix for v(r) to make D ∈ top(r). The antidiagonal elbows come from the pipe added to make s_{0n} ≥ 1: by Theorem 5.10 there is a pipe going from the southwest block to the northeast block, and by Theorem 3.8 this pipe crosses no others. In particular, the southeast corner of each superantidiagonal block in D′ is an elbow. Lemma 5.13 implies the southern elbow row, and its transpose (applied to v(1 + r)^{−1} = v(1 + r′) for the transpose r′ of r) implies the western elbow column. □

Interpreting Proposition 5.15 on lacing diagrams for pipe dreams yields another useful stability statement that we shall apply in Section 6; compare Corollary 4.12.

Corollary 5.16. Let W_{RP}(r) = {w(D) | D ∈ RP(v(r))} be the set of lacing diagrams obtained by decomposition of reduced pipe dreams for v(r) into horizontal strips as in Definition 5.8 and Theorem 5.10. Then W_{RP}(m + r) = {m + w | w ∈ W_{RP}(r)} is obtained by shifting each partial permutation list in W_{RP}(r) up by m.

Proof. Let D ∈ top(r) correspond to D′ ∈ top(1 + r). The miniature top pipe dream D′_{j} in the j^{th} horizontal strip of D′ is one row farther from the bottom row of that strip than D_{j} is from the bottom row in its strip, by Lemma 5.11 and Proposition 5.15. Reading (partial) permutations in each horizontal strip with the southeast corner as the origin, the result follows from Lemma 5.14. □

Section 6. Component formulae

Our goal in this section is to prove that stable limits of double quiver polynomials exist for uniformly increasing ranks (Theorem 6.11), and to deduce from it the various forms of the component formula (Corollary 6.17, Theorem 6.20, and Corollary 6.23). Along the way, we complete the combinatorial characterization of the geometric components in quiver degenerations (Theorem 6.16). Since the proofs gather together most of what we have done, we begin this section with an interlude: a summary of the roles played by the results thus far in the coming sections.

A combinatorial hierarchy. It will be helpful to have in mind a natural ordering of the four formulae, different from the order in which we presented them in the Introduction (which was determined by our proofs). The combinatorial objects indexing the summands in the four formulae can all be described in terms of the set RP(v(r)) of reduced pipe dreams for the Zelevinsky permutation v(r). More precisely, there are four partitions of the set RP(v(r)) such that each type of combinatorial object corresponds to an equivalence class in one of these partitions. The hierarchy is as follows.

1. Pipe formula: equivalence classes are singletons.

2. Tableau formula: equivalence classes are key classes. There is a bijection from pipe dreams in RP(v(r)) to ‘compatible pairs’ (a, i) in which a is a reduced word for the Zelevinsky permutation and i is a ‘compatible sequence’ of positive integers [BB93]. The set of compatible pairs can be partitioned according to the Coxeter–Knuth classes of the reduced words a. The generating function of each equivalence class with respect to the weight of the compatible sequences i is known as a ‘key polynomial’ or ‘Demazure character of type A’ [RS95b]. These polynomials are crucial to our derivation of the tableau formula (Section 7).

3. Component formula: equivalence classes are strip classes, indexed by lacing diagrams. One of our favorite combinatorial results in this paper is the combination of Theorem 5.10 and Corollary 6.18. These say that out of every reduced pipe dream for v(r) comes a minimal length lacing diagram with rank array r. Section 5.2 describes the simple construction, which divides each pipe dream into horizontal strips (block rows). Two pipe dreams are equivalent if they produce the same lacing diagram—that is, if their pipes carry out the same partial permutations in each strip.

4. Ratio formula: only one equivalence class.

Roughly speaking, summing monomials (x − ^{∘}x)^{D∖D_{Hom}} over an equivalence class in one of these four partitions yields a summand in one of the four formulae. This is contentfree for the ratio and pipe formulae. Moreover, we do not explicitly use this idea for the tableau formula, although it is faintly evident in the proof of Proposition 7.13. That proof is also suggestive of the essentially true statement that key equivalence refines lacing equivalence, which is the fundamental (though not explicit) principle behind our derivation of the peelable tableau formula from the component formula. For the component formula itself, summation over a lacing equivalence class is an essential operation, but it only roughly yields a lacing diagram summand. This is the first of many situations where stable limits come to the rescue.

Limits and stabilization. There are two primary aspects to our proof of the ordinary component formula: we must show that

• components in the degeneration Ω_{r}(0) have multiplicity 1, and they are exactly the closures of the orbits through minimal length lacing diagrams with ranks r; and • the summands can be taken to be products of Stanley symmetric functions rather

than products of double Schubert polynomials. The Schubert version for ordinary quiver polynomials follows immediately from the first of the two statements above, by Corollary 4.9. But keep in mind that we need to prove a component formula for double quiver polynomials to be able to deduce the tableau formula for quiver constants, as outlined at the end of the Introduction.

The two aspects itemized above split the proof into two essentially disjoint parts: a geometric argument producing a lower bound, and a combinatorial argument producing an upper bound. The conclusion comes by noticing that the lower bound is visibly at least as high as the upper bound, so that both must be equal. Let us make this vague description more precise, starting with the geometric lower bound.

The mere existence of the flat degeneration in Section 4.2 automatically implies that the ordinary quiver polynomial Q_{r}(x − ^{∘}x) is a sum of double Schubert products 𝔖_{w}(x − ^{∘}x), with positive integer coefficients possibly greater than 1 (Corollary 4.9). This is because the components in the degeneration are products of matrix Schubert varieties, possibly nonreduced a priori (Theorem 4.6). Moreover, it is easy to verify that the orbit through each lacing diagram in W (r) appears with multiplicity at least 1 (Proposition 4.11).

One of the fundamental observations in [BF99] gained by viewing quiver polynomials as ordinary classes of degeneracy loci for vector bundle maps on arbitrary schemes is that nothing vital changes when the ranks of all the bundles are increased by 1, or by any fixed integer m. The key observation for us along these lines is that this stability under uniformly increasing ranks can be proved directly for the total families of quiver locus degenerations— and hence for components in the special fiber, including their scheme structures—by a direct geometric argument (Proposition 4.13). As a consequence, the previous paragraph, which a priori only gives existence of a Schubert polynomial component formula along with lower bounds on its coefficients, in fact proves the same existence and lower bounds for an ordinary Stanley function component formula, simply by taking limits (see Remark 4.15).

The combinatorial argument producing an upper bound is based on the partition of the pipe dreams in RP(v(r)) into strip classes, and it automatically works in the ‘double’ setting, with x and y alphabets. Summing (x − ^{∘}y)^{D∖D_{Hom}} over reduced pipe dreams D with strip equivalence class w yields a polynomial that, while not actually equal to 𝔖_{w}(x − ^{∘}y) (Remark 6.26), agrees with 𝔖_{w}(x − ^{∘}y) in all coefficients on monomials all of whose variables appear “not too late” in the alphabets x^{0}, …, x^{n} and y^{0}, …, y^{n}; see Corollary 6.10. This observation depends subtly on the combinatorics of reduced pipe dreams, and on the symmetry of Q_{r}(x − ^{∘}y) in each of its 2n + 2 alphabets as in Proposition 6.9.

Just as in the geometric argument, we have a stability statement. Here, it says that strip classes are stable under uniform increase in rank (Corollary 5.16). Consequently, summing over all strip classes and taking the limit as the ranks uniformly become infinite yields Theorem 6.11, the “existence and upper bound theorem” for the resulting double quiver functions F_{r}(x − ^{∘}y). That the sum is over strip classes, whose lacing diagrams automatically lie inside W (r) by Theorem 5.10, implies the upper bound: the coefficients on F_{w}(x − ^{∘}y) is 1 if w indexes a strip class, and zero otherwise.

After specializing the strip class component formula in Theorem 6.11 to ^{∘}y = ^{∘}x, the geometric lower bound and the combinatorial upper bound are forced to match. This argument comprises the proof of Theorem 6.16 on the combinatorics and geometry of the components in quiver degenerations, and it also implies the bijection between strip classes and lacing diagrams of minimal length (Corollary 6.18). The stable double component formula (Theorem 6.20) is an immediate consequence.

To convey a better feel for what limits accomplish here, let us summarize the fundamental principle that drives most of our logic. There are a number of geometric, algebraic, and combinatorial objects that we are unable to get a precise handle on directly. These objects include the components of quiver degenerations, as well as their multiplicities; the sum of all monomials associated to pipe dreams in a fixed strip class; and the set of lacing diagrams indexing strip classes for a fixed Zelevinsky permutation. (In an earlier draft, peelable tableaux were on this list, as well; see Corollary 8.21 and the comment following it.) The key realization is that, even though we fail to identify these objects precisely, we can prove their stability under uniformly increasing ranks, and in the limit it becomes possible to identify the desired objects exactly. Hence our failures occurred only because we fixed the ranks to begin with.

Let us note that our failures in fixed finite rank are not always artifacts of our proof technique. For example, the sum of all monomials for pipe dreams in a fixed strip class w really isn’t equal to the Stanley product F_{w} or the Schubert product 𝔖_{w}; it only becomes so in the limit.

6.1. Double Stanley symmetric functions

Symmetric functions are formal power series in infinite alphabets, which may be specialized to polynomials in alphabets of finite size. We prepare conventions for such specializations, since they will be crucial to our line of reasoning.

Convention 6.1. If a polynomial requiring r or more variables for input is evaluated on an alphabet with r′ < r letters, then the remaining r − r′ variables are to be set to zero. Conversely, if a polynomial requiring no more than r variables for input is evaluated on an alphabet of size r′ > r, then the last r′ − r variables are ignored.

Example 6.2. Consider the four bc blocks obtained from the big pipe dream in Exam-

ple 5.4: y_{1}^{2} y_{2}^{2} y_{3}^{2} y_{4}^{2} y_{1}^{1} y_{2}^{1} y_{3}^{1}

x_{1}^{1} · · + + ⋯
x_{2}^{1} ⋯ + · ·
x_{3}^{1} · · · · · · ·
x_{1}^{2} · + · · · · ·
x_{2}^{2} · · · · · · ·
x_{3}^{2} · · · · · · ·
x_{4}^{2} · · · · · · ·

This is a reduced pipe dream for the permutation w = 1253746. The double Schubert polynomial 𝔖_{1253746}(x^{1}, x^{2} − y^{2}, y^{1}) is a sum of ‘monomials’ of the form (x − y), one for each reduced pipe dream D ∈ RP(1253746). Setting x^{2} = y^{1} = 0 in 𝔖_{1253746}(x^{1}, x^{2} − y^{2}, y^{1}) yields 𝔖_{1253746}(x^{1} − y^{2}). In the expression of this polynomial as a sum of ‘monomials’, reduced pipe dreams with crosses outside the upper-left block contribute differently to 𝔖_{w}(x^{1} − y^{2}) than they did to 𝔖_{w}(x^{1}, x^{2} − y^{2}, y^{1}). The pipe dream depicted above, for example, contributes (x_{1}^{1} − y_{3}^{2})(x_{1}^{1} − y_{4}^{2})(x_{2}^{1} − y_{1}^{1})(x_{1}^{2} − y_{2}^{2}) to 𝔖_{w}(x^{1}, x^{2} − y^{2}, y^{1}), but only (x_{1}^{1} − y_{3}^{2})(x_{1}^{1} − y_{4}^{2})(x_{2}^{1})(−y_{2}^{2}) to 𝔖_{w}(x^{1} − y^{2}). □

Suppose that x = x^{0}, x^{1}, …, x^{n} and ^{∘}y = y^{n}, …, y^{1}, y^{0} are two ordered finite lists of infinite alphabets. A sequence p_{m}(x, ^{∘}y) of polynomials in these lists (or any infinite alphabet) is said to converge to a power series p(x, ^{∘}y) if the coefficient on any fixed monomial is eventually constant as a function of m. Equivalently, we say the limit exists, and write

lim_{m→∞} p_{m}(x, ^{∘}y) = p(x, ^{∘}y).

Here, p(x, ^{∘}y) is allowed to be an arbitrary formal sum of monomials with integer coefficients in variables from the union of alphabets in the lists x and ^{∘}y.

Recall from Section 4.4 the notation m + w for nonnegative integers m and partial permutations w.

Proposition 6.3. Given a partial permutation w and an infinite alphabet X, the limit

F_{w}(X) = lim_{m→∞} 𝔖_{m+w}(X)

exists and is symmetric in X. (Only finitely many X variables appear in each 𝔖_{m+w}(X).)

Definition 6.4. The limit F_{w}(X) is called the Stanley symmetric function or stable Schubert polynomial for w. (N.B. In the notation of [Sta84] the permutation w^{−1} is used for indexing instead of w).

Proof. After harmlessly extending w to an honest permutation in the minimal way (which leaves the Schubert polynomials unchanged), see [Mac91, Section VII]. □

Stanley’s original definition of F_{w} was combinatorial, and motivated by the fact that the coefficient of any squarefree monomial in F_{w} is the number of reduced decompositions of the permutation w. The fact that Stanley’s formulation of F_{w} is a stable Schubert polynomial in the above sense follows immediately from the formula for the Schubert polynomial given in [BJS93, Theorem 1.1].

Being a symmetric function, F_{w} can be written as a sum

(6.1)  F_{w}(X) = ∑_{λ} α_{w}^{λ}s_{λ}(X)

of Schur functions s_{λ} with coefficients α_{w}^{λ}, which we call Stanley coefficients. Stanley credits Edelman and Greene [EG87] with the first proof that these coefficients α_{w}^{λ} are nonnegative. There are various descriptions for α_{w}^{λ}, all combinatorial: reduced word tableaux [LS89, EG87], the leaves of shape λ in the transition tree for w [LS85], certain promotion sequences [Hai92], and peelable tableaux [RS95a]. The coefficients α_{w}^{λ} are also special cases of the quiver constants c_{\underline{λ}}(r) from Theorem 1.20 [Buc01a]. We shall prove the converse in Theorem 7.14, below: the quiver constants c_{\underline{λ}}(r) are special cases of Stanley coefficients α_{w}^{λ}.

Proposition 6.5. Given a partial permutation w and infinite alphabets X , Y, the limit

F_{w}(X − Y) = lim_{m→∞} 𝔖_{m+w}(X − Y)

of double Schubert polynomials exists and is symmetric separately in X and Y.

Definition 6.6. The limit F_{w}(X − Y) is the double Stanley symmetric function or stable double Schubert polynomial.

Proof. Again, we may as well assume w is an honest permutation. One may reduce to the “single” stable Schubert case by applying the coproduct formula [Mac91, (6.3)]

𝔖_{w}(X − Y) = ∑_{u,v}(−1)^{ℓ(v)}𝔖_{u}(X)𝔖_{v^{−1}}(Y)

where the sum runs over the pairs of permutations u, v such that vu = w and the sum ℓ(u) + ℓ(v) of the lengths of v and u equals the length ℓ(w) of w. □

For a sequence w = w_{1}, …, w_{n} of partial permutations of size r_{0}×r_{1}, …, r_{n−1}×r_{n}, and two sequences x = x^{0}, …, x^{n} and ^{∘}y = y^{n}, …, y^{0} of infinite alphabets, denote by

𝔖_{w}(x − ^{∘}y) = ∏_{j=1}^{n} 𝔖_{w_{j}}(x_{r}^{j−1}− y_{r}^{j})

the product of double Schubert polynomials in consecutive x and y alphabets. As in Definitions 1.18 and 2.5, only finitely many variables actually appear in 𝔖_{w}(x − ^{∘}y). Write

(6.2)  F_{w}(x − ^{∘}y) = ∏_{j=1}^{n} F_{w_{j}}(x^{j−1}− y^{j})

for the corresponding product of double Stanley symmetric functions, where now all of the alphabets are infinite. If, in (6.2), we need to specify finite alphabets (2.3), when ranks r have been set so that x^{j} and y^{j} have size r_{j}, then we shall write explicitly F_{w}(x_{r} − ^{∘}y_{r}). Keep in mind that the power series in (6.2) remains unchanged when w = (w_{1}, …, w_{n}) is replaced by m + w for m ∈ ℕ.

6.2. Double quiver functions

Stanley symmetric functions arise in our context by taking limits of double quiver polynomials via the pipe formula, Theorem 5.5. Each reduced pipe dream D for the Zelevinsky permutation v(r) determines a polynomial 𝔖_{w(D)}(x − ^{∘}y) by Definition 5.8, to which D contributes a “monomial”, as in Section 5.1. Here, in Proposition 6.9, we shall see that each reduced pipe dream for v(r) can contribute a different monomial by way of a crucial symmetry, as follows.

Definition 6.7. The reverse monomial associated to a d×d pipe dream D is the product (x̃_{r} − ỹ_{r})^{D} over all entries in D of (x̃_{+} − ỹ_{+}), where the variable x̃_{+} sits at the left end of the row containing after reversing each of the n + 1 alphabets in x_{r}, and ỹ_{+} sits atop the column containing after reversing each of the n + 1 alphabets in ^{∘}y_{r}.

Example 6.8. Reversing each of the row and column alphabets in Example 5.4 gives

\begin{aligned}& y_{3}^{3} & y_{2}^{3} & y_{1}^{3} & y_{4}^{2} & y_{3}^{2} & y_{2}^{2} & y_{1}^{2} & y_{3}^{1} & y_{2}^{1} & y_{1}^{1} & y_{2}^{0} & y_{1}^{0} \\ x_{2}^{0} & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & · & · & + & · & · \\ x_{1}^{0} & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & · & · & · & · & · \\ x_{3}^{1} & ∗ & ∗ & ∗ & · & · & + & + & · & · & · & · & · \\ x_{2}^{1} & ∗ & ∗ & ∗ & · & · & · & · & + & · & · & · & · \\ x_{1}^{1} & ∗ & ∗ & ∗ & · & · & · & · & · & · & · & · & · \\ x_{4}^{2} & · & · & + & · & + & · & · & · & · & · & · & · \\ x_{3}^{2} & · & · & · & · & · & · & · & · & · & · & · & · \\ x_{2}^{2} & · & + & · & · & · & · & · & · & · & · & · & · \\ x_{1}^{2} & · & · & · & · & · & · & · & · & · & · & · & · \\ x_{3}^{3} & · & · & · & · & · & · & · & · & · & · & · & · \\ x_{2}^{3} & · & · & · & · & · & · & · & · & · & · & · & · \\ x_{1}^{3} & · & · & · & · & · & · & · & · & · & · & · & · \end{aligned} = \begin{aligned}& ḋ_{3} & ḋ_{2} & ḋ_{1} & ċ_{4} & ċ_{3} & ċ_{2} & ċ_{1} & ḃ_{3} & ḃ_{2} & ḃ_{1} & ȧ_{2} & ȧ_{1} \\ a_{2} & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & · & · & + & · & · \\ a_{1} & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & ∗ & · & · & · & · & · \\ b_{3} & ∗ & ∗ & ∗ & · & · & + & + & · & · & · & · & · \\ b_{2} & ∗ & ∗ & ∗ & · & · & · & · & + & · & · & · & · \\ b_{1} & ∗ & ∗ & ∗ & · & · & · & · & · & · & · & · & · \\ c_{4} & · & · & + & · & + & · & · & · & · & · & · & · \\ c_{3} & · & · & · & · & · & · & · & · & · & · & · & · \\ c_{2} & · & + & · & · & · & · & · & · & · & · & · & · \\ c_{1} & · & · & · & · & · & · & · & · & · & · & · & · \\ d_{3} & · & · & · & · & · & · & · & · & · & · & · & · \\ d_{2} & · & · & · & · & · & · & · & · & · & · & · & · \\ d_{1} & · & · & · & · & · & · & · & · & · & · & · & · \end{aligned}

This reduced pipe dream contributes the reverse monomial

(a_{2} − ḃ_{1})(b_{3} − ċ_{2})(b_{3} − ċ_{1})(b_{2} − ḃ_{3})(c_{4} − ḋ_{1})(c_{4} − ċ_{3})(c_{2} − ḋ_{2})

to the double quiver polynomial Q_{r}(a, b, c, d − ḋ, ċ, ḃ, ȧ). Reversing the alphabets in the argument of this polynomial leaves it invariant, because by Proposition 6.9 this polynomial already equals the sum of all reverse monomials from reduced pipe dreams for v(r). □

In conjunction with Corollary 6.10, the next observation will allow us to take limits of double quiver polynomials effectively, via Proposition 6.5.

Proposition 6.9. The double quiver polynomial Q_{r}(x − ^{∘}y) is symmetric separately in each of the alphabets x_{r}^{i} as well as in each of the alphabets y_{r}^{j}. In particular,

Q_{r}(x − ^{∘}y) = ∑_{D∈RP(v(r))}(x̃_{r} − ỹ_{r})^{D∖D_{Hom}}.

where (x̃_{r}−ỹ_{r})^{D∖D_{Hom}} is the reverse monomial associated to the d×d pipe dream D∖D_{Hom}.

Proof. Matrix Schubert varieties for Zelevinsky permutations are preserved by the action of block diagonal matrices—including block diagonal permutation matrices—on the right and left. Now apply the symmetry to Theorem 5.5. □

Next we record an important consequence of Proposition 5.15.

Corollary 6.10. There is a fixed integer ℓ, independent of m, such that setting the last ℓ variables to zero in every finite alphabet from the lists x_{m+r} and ^{∘}y_{m+r} kills the reverse monomial for every reduced pipe dream D ∈ RP(v(m+r)) containing at least one cross in an antidiagonal block.

Proof. Suppose D is a reduced pipe dream for v(m + r). Divide D into horizontal strips (block rows) as in Section 5.3, and consider the reduced pipe dream D′ in the same chute class as D that consists of a top pipe dream in each strip. The miniature top pipe dream in horizontal strip j has an eastern half D′_{j} in the j^{th} antidiagonal block. Inside the j^{th} antidiagonal block, consider the highest antidiagonal A_{j} above which every in D′_{j} lies. Thus A_{j} bounds an isosceles right triangular region ▽_{j}(D) in the northwest corner of the j^{th} antidiagonal block that contains every in D′_{j}.

We claim that ▽_{j}(D) in fact contains every of D itself that lies in the j^{th} antidiagonal block. Indeed, suppose that D contains a in the j^{th} antidiagonal block. Using (any sequence of) upward chute moves to get from D to D′, the antidiagonal we are considering must end up inside ▽_{j}(D), because chuting any up pushes it north and east. Our claim follows because upward chutes push each at least as far east as north.

Now define ℓ(D) to be the maximum of the leg lengths of the isosceles right triangles ▽_{j}(D). Chuting up as in the previous paragraph, both the row variable x̃_{+} and the column variable ỹ_{+} of each antidiagonal in D must lie within ℓ(D) of the end of its finite alphabet. Therefore, pick ℓ to be the maximum of the numbers ℓ(D) for reduced pipe dreams D ∈ RP(v(m + r)). The key (and final) point is that ℓ does not depend on m for m ≥ 0, by Proposition 5.15. □

The main result of this section concerning double quiver polynomials Q_{r}(x−^{∘}y) says that limits exist as the ranks r get replaced by m + r for m → ∞. We consider this an algebraic rather than a combinatorial result (the latter being Theorem 6.20), since we are unable at this stage to identify the set W_{RP}(r) from Corollary 5.16 in a satisfactory way. Note that W_{RP}(r) is a set of distinct partial permutation lists—not a multiset—contained in the set W (r) of minimum length lacing diagrams with ranks r, by Theorem 5.10. We shall see in Corollary 6.18 that in fact W_{RP}(r) = W (r); but until then it is important for the proofs to distinguish W_{RP}(r) from W (r).

Theorem 6.11. The limit of double quiver polynomials Q_{m+r}(x−^{∘}y) for m approaching ∞ exists and equals the multiplicity-free sum

F_{r}(x − ^{∘}y) := lim_{m→∞} Q_{m+r}(x − ^{∘}y) = ∑_{w∈W_{RP}(r)} F_{w}(x − ^{∘}y)

of products of double Stanley symmetric functions. The limit power series F_{r}(x − ^{∘}y) is symmetric separately in each of the 2n + 2 infinite alphabets x^{0}, …, x^{n} and y^{n}, …, y^{0}.

Definition 6.12. F_{r}(x − ^{∘}y) is called the double quiver function for the rank array r.

Proof. For all m ≥ 0, the set W_{RP}(m + r) is obtained from W_{RP}(r) by replacing each permutation list w with m + w, by Corollary 5.16. For each partial permutation list w ∈ W_{RP}(r), consider the set RP_{w}(v(m + r)) of reduced pipe dreams D for v(m + r) whose lacing diagrams w(D) equal m+w. The sum of all reverse monomials for pipe dreams D ∈ RP_{w}(v(m+r)) is a product of polynomials, each factor coming from the miniature pipe dreams in a single horizontal strip. The double quiver polynomial Q_{m+r}(x − ^{∘}y) is the sum of these products, the sum being over lacing diagrams w ∈ W_{RP}(r). A similar statement can be made after setting the last ℓ variables in each alphabet to zero as in Corollary 6.10, although now Q_{r}(x − ^{∘}y) and the polynomials for each lacing diagram w ∈ W_{RP}(r) have changed a little (but only in those terms involving a variable near the end of some alphabet).

We claim that now, after setting the last few variables in each alphabet to zero, the j^{th} polynomial in the product for any fixed diagram w equals the result of setting the last few variables (at most ℓ) to zero in the honest double Schubert polynomial 𝔖_{m+w_{j}}(x^{j−1} − y^{j}). For this, note that every reduced pipe dream D for v(r) contributing a nonzero reverse monomial now has locations only in superantidiagonal blocks. Hence we can read the restriction of D to the j^{th} superantidiagonal block as a reduced pipe dream with its southeast corner as the origin. Since the variables in this one-block rectangle that we have set to zero are far from the southeast corner, our claim is proved.

Taking the limit as m → ∞ produces the desired product of double Stanley symmetric functions by Proposition 6.5. □

The limit in Theorem 6.11 stabilizes in the following strong sense. The analogous stabilization for the limit defining Stanley symmetric functions from Schubert polynomials (Proposition 6.3) almost never occurs, since the finite variable specialization is symmetric, while Schubert polynomials rarely are, even considering only variables that actually occur.

Corollary 6.13. For all m ≫ 0, the double quiver polynomial Q_{m+r}(x − ^{∘}y) actually equals the specialization F_{r}(x_{m+r} − ^{∘}y_{m+r}) of the double quiver function.

Proof. The double quiver polynomial Q_{m+r}(x−^{∘}y) has fixed degree and is symmetric in each of its finite alphabets for all m (Proposition 6.9). Hence we need only show that for any fixed monomial of degree at most deg(Q_{m+r}(x − ^{∘}y)), applying a symmetry yields a monomial whose coefficient equals the coefficient on the corresponding monomial in F_{r}(x − ^{∘}y). This follows immediately from Theorem 6.11, because for m ≫ 0, all monomials involving variables whose lower indices are at most deg(Q_{m+r}(x−^{∘}y)) have the correct coefficients. □

The previous result led us to suspect that the ‘≫’ sign may be replaced by ‘≥’.

Conjecture 6.14. Q_{m+r}(x − ^{∘}y) = F_{r}(x_{m+r} − ^{∘}y_{m+r}) for all m ≥ 0.

After we made Conjecture 6.14 in the first draft of this paper, A. Buch responded by proving it in [Buc03]. We have no need for the full strength of Conjecture 6.14 in this paper, since Corollary 6.13 suffices; see Proposition 7.13, below, and the last line of its proof. We shall comment on related issues in Remarks 6.24, 6.25, and 6.26.

We shall see later, after a number of intermediate steps leading to Theorem 7.10, that F_{r}(x − ^{∘}y) is equal to the power series obtained by replacing the second argument ^{∘}x of the Buch–Fulton power series ∑_{\underline{λ}} c_{\underline{λ}}(r)s_{\underline{λ}}(x−^{∘}x) with ^{∘}y, or equivalently by the appropriate specialization of the abstract symmetric functions P_{r} in [Buc01a] to pairs of infinite alphabets in each factor. As a consequence, the double quiver functions turn out to equal the sums of lowest-degree terms in the power series P_{r} defined by Buch in [Buc02, Section 4]. But until Theorem 7.10, the term ‘double quiver function’ will refer to the limit in Theorem 6.11.

6.3. All components are lacing diagram orbit closures

Recall from Theorem 2.9 that the quiver polynomial Q_{r}(x − ^{∘}x) is obtained from the double quiver polynomial Q_{r}(x − ^{∘}y) by setting y^{j} = x^{j} for all j, so ^{∘}y = ^{∘}x. Here, we shall need to specialize products F_{w}(x − ^{∘}y) and 𝔖_{w}(x − ^{∘}y) of double Schubert and Stanley symmetric functions by setting ^{∘}y = ^{∘}x, to get F_{w}(x − ^{∘}x) and 𝔖_{w}(x − ^{∘}x). Note that the alphabet x^{j} obtained by specializing y^{j} never interferes in a catastrophic cancelative manner with the original alphabet x^{j}, because for any lacing diagram w, the j^{th} factors of 𝔖_{w}(x−^{∘}x) and F_{w}(x − ^{∘}x) involve the distinct alphabets x^{j−1} and x^{j}.

Lemma 6.15. Any nonempty positive sum of products F_{w}(x − ^{∘}x) of double Stanley symmetric functions in differences of consecutive alphabets is nonzero.

Proof. One may use the double version of (6.1) to write each term F_{w}(x−^{∘}x) as a nonnegative sum of products of Schur functions in differences of consecutive alphabets. Such products are linearly independent by the remark in [BF99, Section 2.2]. □

Now we can finally identify the components of quiver degenerations.

Theorem 6.16. The components of the quiver degeneration Ω_{r}(0) are exactly the orbit closures O(w) for w ∈ W (r). Furthermore, Ω_{r}(0) is generically reduced along each component O(w), so its multiplicity c_{w}(r) there equals 1.

Proof. Applying Corollary 4.9 to Q_{m+r}(x − ^{∘}x) instead of Q_{r}(x − ^{∘}x) yields

Q_{m+r}(x − ^{∘}x) = ∑_{w∈Hom} c_{w}(r)𝔖_{m+w}(x − ^{∘}x)

by Proposition 4.13. Taking the limit as m → ∞ in the above equation, we find that

(6.3)  F_{r}(x − ^{∘}x) = ∑_{w∈Hom} c_{w}(r)F_{w}(x − ^{∘}x),

the right side by Proposition 6.5 and the left by specializing Theorem 6.11 to ^{∘}y = ^{∘}x.

On the other hand, the right side of Theorem 6.11 becomes a subsum of (6.3) after specializing to ^{∘}y = ^{∘}x, by Theorem 5.10 and Proposition 4.11. The remaining terms from (6.3) sum to zero, so there can be no remaining terms by Lemma 6.15. We conclude that in fact W_{RP}(r) = W (r) and c_{w}(r) = 1 for all w ∈ W (r). □

Corollary 6.17 (Component formula for quiver polynomials—Schubert version). The quiver polynomial Q_{r}(x − ^{∘}x) equals the sum

Q_{r}(x − ^{∘}x) = ∑_{w∈W (r)} 𝔖_{w}(x − ^{∘}x)

of products of double Schubert polynomials (in consecutive alphabets x = x^{0}, …, x^{n}) indexed by minimum length lacing diagrams with rank array r.

We have chosen to present this consequence of Theorem 6.16 and Corollary 4.9 now, even though it is a specialization of the stable double component formula in the next section, to emphasize that it has a direct geometric interpretation: the right-hand side is the sum of the equivariant cohomology classes of components in the quiver degeneration. The double formula to come in Theorem 6.20 currently lacks such a geometric interpretation, and moreover the reason why it implies Corollary 6.17 is somewhat subtle; see Corollary 6.23 and Remark 6.26. The subtlety in this argument is related to the fact, discussed in Remark 6.26, that replacing ^{∘}x with ^{∘}y in Corollary 6.17 always yields a false statement whenever there is more than one term on the right-hand side.

In the course of proving Theorem 6.16, we reached a notable combinatorial result.

Corollary 6.18. Every minimal length lacing diagram for a rank array r occurs as the lacing diagram derived from a reduced pipe dream for the Zelevinsky permutation v(r):

W (r) = W_{RP}(r).

Remark 6.19. Theorem 5.10 and Corollary 6.18 give another explanation (and proof) for Theorem 3.8. That laces of w ∈ W (r) can cross at most once is exactly the statement that pipes in reduced pipe dreams for v(r) cross at most once. That laces starting or ending in the same column do not cross at all follows from the fact that Zelevinsky permutations and their inverses have no descents within block rows.

6.4. Stable double component formula

Theorem 6.20. The double quiver function can be expressed as the sum

F_{r}(x − ^{∘}y) = ∑_{w∈W (r)} F_{w}(x − ^{∘}y)

of products of double Stanley symmetric functions indexed by minimal lacing diagrams w.

Proof. Using Corollary 6.18, we can replace the sum in Theorem 6.11 over W_{RP}(r) by one over W (r). □

Remark 6.21. Definition 5.8 reads pipes as if they flow northeast to southwest (compare Definition 5.1) because our proof of Theorem 6.11 requires that we read antidiagonal blocks in Zelevinsky pipe dreams as miniature pipe dreams with their southeast corners as their origins. This method suggests a direct combinatorial proof of Corollary 6.18, without appealing to the geometric lower bound on multiplicities of components in quiver degenerations afforded by Corollary 4.9. This idea has since been carried out in [Yon03]. Thus the component formula in Theorem 6.20 can be proved in a purely combinatorial way using Theorem 6.11, if one assumes the ratio formula (whose proof requires geometry).

Example 6.22. In the case of Fulton rank conditions, the ranks r_{0}, r_{1}, r_{2}, r_{3}, … equal 1, 2, 3, …, n − 1, n, n, n − 1, …, 3, 2, 1 and r = r_{w} is specified by a permutation w in S_{n+1}. Theorem 6.20 contains a combinatorial formula found independently in [BKTY02]. This is because the Zelevinsky permutation v(r_{w}) has as many diagonal × entries as will fit in superantidiagonal blocks, and the southeast corner (last n+1 block rows and block columns) is a block version of the permutation w itself (rotated by 180^{∘}). See [Yon03] for details. □

Corollary 6.23 (Component formula for quiver polynomials—Stanley version). The quiver polynomial can be expressed as the specialization

Q_{r}(x − ^{∘}x) = F_{r}(x_{r} − ^{∘}x_{r}) = ∑_{w∈W (r)} F_{w}(x_{r} − ^{∘}x_{r})

of the quiver function F_{r}(x − ^{∘}x) to a sequence x_{r} of finite alphabets of size r_{0}, …, r_{n}.

Proof. Since Theorem 1.20 holds for any ranks r, it holds in particular for m + r when m ≫ 0. Moreover, it is shown in [BF99] that

(6.4)  c_{\underline{λ}}(m + r) = c_{\underline{λ}}(r)

for all \underline{λ} and m ≥ 0. Specializing Theorem 6.11 to ^{∘}y = ^{∘}x, we may take the limit there with ∑ c_{\underline{λ}}(r)s_{\underline{λ}}(x_{m+r} − ^{∘}x_{m+r}) in place of Q_{m+r}(x − ^{∘}x), by Theorem 2.9 and Theorem 1.20. We conclude that

(6.5)  F_{r}(x − ^{∘}x) = ∑_{\underline{λ}} c_{\underline{λ}}(r)s_{\underline{λ}}(x − ^{∘}x)

as power series in the ring of functions that are symmetric in each of the infinite sets x^{0}, …, x^{n} of variables. Specializing each x^{j} in (6.5) to have r_{j} nonzero variables yields Q_{r}(x − ^{∘}x) by Theorem 1.20, and it yields ∑_{w∈W (r)} F_{w}(x_{r} − ^{∘}x_{r}) by Theorem 6.20. □

Remark 6.24. As a consequence of Corollary 6.23, there is no need to define ‘stable quiver polynomials’, just as there is no need to define ‘stable Schur polynomials’. The fact that Conjecture 6.14 is actually true, as proved in [Buc03], means that we are correct to avoid the term ‘stable double quiver polynomial’ in favor of the more apt ‘double quiver function’.

Remark 6.25. We chose our definition of the double quiver polynomial Q_{r}(x − ^{∘}y) via the ratio formula from among four possibilities that we considered, three of which turned out to be pairwise distinct. In hindsight we could equally well have chosen the candidate obtained by replacing ^{∘}x with ^{∘}y in the expression ∑ c_{\underline{λ}}(r)s_{\underline{λ}}(x_{r} − ^{∘}x_{r}) from Theorem 1.20, or equivalently by (6.5), in the expression ∑ F_{w}(x_{r} − ^{∘}x_{r}) from Corollary 6.23. However, it was the algebraic and combinatorial properties derived from the ratio formula that made Q_{r}(x − ^{∘}y) useful to us. Moreover, our methods did not prove the fact that this other candidate is equal to our double quiver polynomials; this is the content of Conjecture 6.14, which we now know to hold only because of its subsequent proof by Buch [Buc03].

One of our other two candidates for double quiver polynomial was obtained by replacing ^{∘}x with ^{∘}y on the right-hand side of Corollary 6.17. As we explain in Remark 6.26, below, this turned out to be less natural than we had expected from seeing the geometric degeneration of quiver loci to unions of products of matrix Schubert varieties.

The remaining candidate for double quiver polynomial was actually an ordinary quiver polynomial, but for a quiver of twice the length as the original. Given a lacing diagram w with rank array r, the “doubled” ranks r^{2} were defined as the rank array of a “doubled” lacing diagram w^{2}. To get w^{2} from w, elongate each column of dots in w to a “ladder” with two adjacent columns (of the same height) and horizontal rungs connecting them. Replacing the even-indexed alphabets in Q_{r^{2}}(x − ^{∘}x) with y alphabets yielded our final candidate.

It is readily verified that the three candidates for double quiver polynomial differ when n = 3, the dimension vector is (1, 2, 1), and the Zelevinsky permutation is 2143 ∈ S_{4}.

Remark 6.26. We now have formulae for quiver polynomials Q_{r}(x − ^{∘}x) in terms of • Stanley symmetric functions F_{w}(x_{r} − ^{∘}x_{r}) in finite alphabets, and

• Schubert polynomials 𝔖_{w}(x − ^{∘}x), which a priori involve finite alphabets. We also have a formula for double quiver functions F_{r}(x − ^{∘}y) in terms of double Stanley symmetric functions F_{w}(x − ^{∘}y), in infinite alphabets. However, the double quiver polynomial Q_{r}(x − ^{∘}y) never equals the sum ∑ 𝔖_{w}(x − ^{∘}y) of double Schubert polynomials for minimum length lacing diagrams w ∈ W (r), unless the sum only has one term, even though

• setting ^{∘}y = ^{∘}x in this sum yields Corollary 6.17, and

• taking limits for uniformly growing ranks yields Theorem 6.20. This failure does not disappear by restricting to ranks m + r for m ≫ 0, either. Indeed, 𝔖_{v(Hom)}(x − ^{∘}y)𝔖_{w}(x − ^{∘}y) equals the double Schubert polynomial 𝔖\overline{w}(x − ^{∘}y) for a certain permutation \overline{w} ∈ S_{d} constructed from w. Hence the linear independence of double Schubert polynomials prevents a direct double generalization of Corollary 6.17, after multiplying such a doubled version by 𝔖_{v(Hom)}(x − ^{∘}y). We view this as evidence for the naturality of double quiver polynomials as we defined them (rather than as ∑_{w∈W (r)}𝔖_{w}(x − ^{∘}y), for example), and even stronger evidence for the naturality of their limits, the double quiver functions. The naturality of double quiver functions also follows from their role in [BF99], where the double quiver functions should be viewed as functionals on lists of bundles that take E_{0}, …, E_{n} to the degeneracy locus formula. The recent proof of our Conjecture 6.14 in [Buc03] cements our belief in the naturality of double quiver polynomials.

The permutation \overline{w} is constructed as follows. Let w′ be the d × d partial permutation matrix obtained by placing each w_{i} in the northwest corner of the superantidiagonal block in row i − 1. Then w′ can be completed uniquely to a permutation \overline{w} ∈ S_{d} by adding nonzero entries in or below the block antidiagonal in such a way that all cells in the diagram of \overline{w} lie in or above the block superantidiagonal.

Geometrically, 𝔖_{v(Hom)}(x−^{∘}y)𝔖_{w}(x−^{∘}y) equals the double Schubert polynomial 𝔖\overline{w}(x−^{∘}y) because of a direct connection between the orbit closure O(w) inside Hom and the matrix Schubert variety \overline{X}\overline{w} inside M_{d}. Moreover, it can be shown that there is a flat (but not Gröbner) degeneration of the matrix Schubert variety \overline{X}_{v(r)} to a generically reduced union \overline{X}_{v(r)}(0) of matrix Schubert varieties for the permutations \overline{w} associated to lacing diagrams w ∈ W (r). However, this flat family is only equivariant for the multigrading by ℤ^{d}, not ℤ^{2d}.

Section 7. Quiver constants

Using the component formula for double quiver functions, we show that the quiver constants c_{\underline{λ}}(r) arise as coefficients in the expansion of the Zelevinsky Schubert polynomial 𝔖_{v(r)} into Demazure characters, and consequently the quiver constants c_{\underline{λ}}(r) equal the corresponding Stanley coefficients α_{w}^{λ}. Using the interpretation of Stanley coefficients as enumerating peelable tableaux [RS95a, RS98], we deduce a combinatorial formula for quiver constants.

7.1. Demazure characters

We recall here the type A_{∞} Demazure characters [Dem74] following [Mac91]. Using the ordinary divided difference ∂_{i} from (2.1), the Demazure operator π_{i}, also called the isobaric divided difference, is the linear operator on ℤ[x_{1}, x_{2}, …] defined by

π_{i}f = ∂_{i}(x_{i}f).

Let σ_{i} be the transposition that exchanges x_{i} and x_{i+1}. The operator π_{i} is idempotent and, like ∂_{i}, commutes with multiplication by any σ_{i}-symmetric polynomial:

∂_{i}(fg) = f∂_{i}(g) and π_{i}(fg) = fπ_{i}(g) if σ_{i}f = f. (7.1)

Since π_{i}1 = 1 it follows that

(7.2)  π_{i}f = f  if σ_{i}f = f.

Given a permutation w ∈ ⋃_{n} S_{n}, let w = σ_{i_{1}} ⋯ σ_{i_{ℓ}} be any reduced decomposition of w, that is, a factorization of w into a minimum number of simple reflections σ_{i}. Define the operator π_{w} = π_{i_{1}} ⋯ π_{i_{ℓ}}. It is independent of the factorization.

Let β = (β_{1}, …, β_{k}) ∈ ℕ^{k} be a sequence of nonnegative integers, and β_{+} the sequence obtained by sorting β into weakly decreasing order. If w ∈ S_{k} is the shortest permutation satisfying β = wβ_{+}, then define the Demazure character κ_{β} ∈ ℤ[x_{1}, …, x_{k}] by

κ_{β} = π_{w}x^{β_{+}}.

For a partition λ ∈ ℕ^{k} and w_{0}^{(k)} ∈ S_{k} the longest permutation, we have

(7.3)  κ_{λ} = x^{λ}
(7.4)  κ_{w_{0}^{(k)}λ} = s_{λ}(x_{1}, x_{2}, …, x_{k}).

Alternatively one may give the recursive definition

(7.5)  κ_{β} = \begin{cases}x^{β} & if β_{1} ≥ β_{2} ≥ ⋯ ≥ β_{k} \\ π_{i}κ_{σ_{i}β} & if β_{i} < β_{i+1}. \end{cases}

It can be shown that

(7.6)  π_{i}κ_{β} = \begin{cases}κ_{β} & if β_{i} ≤ β_{i+1} \\ κ_{σ_{i}β} & if β_{i} ≥ β_{i+1}. \end{cases}

Remark 7.1. The Demazure characters {κ_{β} | β ∈ ℕ^{k}} form a ℤ-basis of the polynomial ring ℤ[x_{1}, x_{2}, …, x_{k}] since

κ_{β} = x^{β} + terms lower in reverse lexicographic order.

Demazure characters interpolate between dominant monomials and Schur polynomials. They have a tableau realization, due to Lascoux and Schützenberger [LS90]. Before describing it, we review some conventions regarding tableaux; for more background and unexplained terminology, see [Ful97].

Our (semistandard) Young tableaux will be drawn with origin at the northwest corner, weakly increasing along each row, and strictly increasing down each column. A word is a sequence of positive integers called “letters”. A column word is one that strictly decreases. In what follows, each tableau t is identified with its column reading word t_{1}t_{2} ⋯, where t_{i} is the column word obtained by reading the letters upward in the i^{th} column of t. Given a word u, denote by [u] the unique tableau equivalent to u under the Knuth-equivalence relations on words: acb ∼ cab for a ≤ b < c and bac ∼ bca for a < b ≤ c [Ful97, Section 2.1].

The weight of a tableau t is the sequence wt(t) = (β_{1}, β_{2}, …), where β_{j} is the number of entries equal to j in t. For each composition β = (β_{1}, β_{2}, …), by which we mean a sequence of nonnegative integers that is eventually zero, the key tableau key(β) of weight β is the unique tableau of shape β_{+} and weight β.

Suppose t has shape λ, so that its j^{th} column t_{j} has length λ′_{j} for 1 ≤ j ≤ r = λ_{1}, where λ′ is the partition conjugate to λ. Given any reordering γ = (γ_{1}, …, γ_{r}) of these column lengths, there is a unique word u = u(t, γ) with a factorization u = u_{1}u_{2} ⋯ u_{r} into column words u_{j} of length γ_{j}, such that [u] = t. Such a word is called frank in [LS90].

The leftmost column factor u_{1} in the word u(t, γ) actually depends only on t and γ_{1}. Define lc_{j}(t) = u_{1} to be the j^{th} left column word of t when γ_{1} equals the length λ′_{j} of the j^{th} column of t. Gathering together all left column words produces the left key of t, which is the tableau K_{−}(t) whose j^{th} column is given by lc_{j}(t) for j = 1, …, r.

Similarly, the rightmost column factor u_{r} in u(t, γ) depends only on t and γ_{r}. Define rc_{j}(t) = u_{r} to be the j^{th} right column word of t when γ_{r} = λ′_{j}, and construct the tableau K_{+}(t), called the the right key of t, whose j^{th} column is given by rc_{j}(t) for j = 1, …, r. The left and right keys of t are key tableaux of the same shape as t.

For tableaux s and t of the same shape, write s ≤ t to mean that every entry of s is less than or equal to the corresponding entry of t. In [LS90], the equation

(7.7)  κ_{β} = ∑_{K_{+}(t)≤key(β)} x^{wt(t)}

is given as a formula for the Demazure character. This implies two positivity properties:

(7.8)  κ_{β} ∈ ℕ[x] for all compositions β, and
(7.9)  κ_{wλ} − κ_{vλ} ∈ ℕ[x] if v ≤ w and λ is a partition.

It is well-known that

(7.10)  s_{λ}(x) = ∑_{tableaux t ; shape(t)=λ} x^{wt(t)}

is a formula for the Schur function s_{λ} in an infinite alphabet x. From (7.7) and (7.10) it follows that

(7.11)  s_{β_{+}} = lim_{m→∞} κ_{(0^{m},β)},

where (0^{m}, β) is the composition obtained from β by prepending m zeros.

7.2. Schubert polynomials as sums of Demazure characters

We now recall the expansion of a Schubert polynomial as a positive sum of Demazure characters and deduce some special properties of this expansion for Zelevinsky permutations.

Theorem 7.2 ([LS89],[RS95a, Theorem 29]). For any (partial) permutation w, there is a multiset M(w) of compositions such that

𝔖_{w}(x) = ∑_{β∈M(w)} κ_{β}.

The Schubert polynomial 𝔖_{w}(x) here is obtained by setting ^{∘}y = 0 in the double Schubert polynomial 𝔖_{w}(x − ^{∘}y) from Section 2.1. The multiset M(w) has been described explicitly using reduced word tableaux in [LS89] and peelable tableaux in [RS95a]. (We shall define the latter in Section 7.4.) The expansion in Theorem 7.2 is a refinement of (6.1), in the sense that taking limits in Theorem 7.2, expresses the Stanley coefficient α_{w}^{λ} in terms of M(w):

Corollary 7.3. The coefficient α_{w}^{λ} in the expansion of a Stanley symmetric function in Schur functions is the number of compositions β ∈ M(w) such that β_{+} = λ.

Proof. By the definition of the Stanley symmetric function and (7.11), it is enough to show that there is a bijection M(w) ↦ M(1+w) given by β ↦ (0, β). Supposing that w lies in S_{k}, the definition of 𝔖_{w} in Section 2.1 says that 𝔖_{w}(x) = ∂_{w′}𝔖_{w_{0}^{(k)}}(x), where w′ = w^{−1}w_{0}^{(k)}

and w_{0}^{(k)} ∈ S_{k} is the longest permutation. Note that

1 + w = σ_{1}σ_{2} ⋯ σ_{k}wσ_{k} ⋯ σ_{2}σ_{1},

and that w_{0}^{(k+1)} = σ_{1}σ_{2} ⋯ σ_{i}w_{0}^{(k)}. Also, in the expression (1 + w)^{−1}w_{0}^{(k+1)} = σ_{1}σ_{2} ⋯ σ_{k}w′, multiplying by each reflection σ_{i} lengthens the permutation. It follows that

𝔖_{1+w} = ∂_{(1+w)^{−1}w_{0}^{(k+1)}}(𝔖_{w_{0}^{(k+1)}}(x))
= ∂_{σ_{1}σ_{2}···σ_{k}w′}(𝔖_{w_{0}^{(k+1)}}(x))
= ∂_{1}∂_{2} ⋯ ∂_{k}∂_{w′}(x_{1} ⋯ x_{k} · 𝔖_{w_{0}^{(k)}}(x))
= ∂_{1}x_{1}∂_{2}x_{2} ⋯ ∂_{k}x_{k}∂_{w′}(𝔖_{w_{0}^{(k)}}(x))
= π_{1} ⋯ π_{k}𝔖_{w}

using the definitions, (7.1), and (2.1).

Let β be a composition in M(w). Since w lies in the symmetric group S_{k} on k letters, the Schubert polynomial 𝔖_{w}(x) involves at most the variables x_{1}, …, x_{k−1}. Hence only the first k − 1 parts of β can be nonzero. By (7.6), π_{1} ⋯ π_{k}κ_{β} = κ_{(0,β)}, as a zero in the (k + 1)^{st} position is swapped to the first position. Since the Demazure characters form a basis (Remark 7.1), the map β ↦ (0, β) is a bijection M(w) ↦ M(1 + w). □

We now recall the dominance bounds on the shapes that occur in the Schur function expansion of Stanley symmetric functions. For any diagram D, by which we mean a set of pairs of positive integers, let D ↑ be the diagram obtained from D by top-justifying the cells in each column. Similarly, let D ← be obtained by left-justifying each row of D. The dominance partial order on partitions of the same size, written as λ ⊵ μ, is defined by the condition that λ_{1} + ⋯ + λ_{i} ≥ μ_{1} + ⋯ + μ_{i} for all i.

Proposition 7.4 ([Sta84]). If the Stanley coefficient α_{w}^{λ} is nonzero, then

D(w)←↑ ⊴ λ ⊴ D(w)↑← .

We require two more lemmata.

Lemma 7.5. If w(i) < w(i + 1) then β_{i} ≤ β_{i+1} for every composition β ∈ M(w).

Proof. Suppose w(i) < w(i + 1). Since σ_{i} ∘ ∂_{i} = ∂_{i}, it follows from the definition of Schubert polynomial that 𝔖_{w}(x) is σ_{i}-invariant. Applying π_{i} to Theorem 7.2 and using (7.2) along with (7.6), it follows that ∑_{β∈M(w)} κ_{β} = ∑_{β∈M(w)} κ_{β′}, where β′ = β if β_{i} ≤ β_{i+1} and β′ = σ_{i}β otherwise. But κ_{β′} − κ_{β} is a polynomial with nonnegative integer coefficients by (7.9). It follows that β′ = β and hence that β_{i} ≤ β_{i+1} for all β ∈ M(w). □

Lemma 7.6. Suppose γ is a composition such that x^{γ} divides 𝔖_{w}(x). Then γ ≤ β for all compositions β ∈ M(w).

Proof. Suppose x^{γ} divides 𝔖_{w}(x). Then x^{γ} divides the sum of Demazure characters on the right-hand side of Theorem 7.2. By (7.8) it follows that x^{γ} divides every monomial of every Demazure character κ_{β} for β ∈ M(w). By Remark 7.1 the reverse lexicographic leading monomial in κ_{β} is x^{β}, so γ ≤ β. □

We now apply the above results to Zelevinsky permutations.

Proposition 7.7. For a fixed dimension vector (r_{0}, r_{1}, …, r_{n}), identify the diagrams D_{Hom} and D(Ω_{0}) from Definition 1.10 with partitions of those shapes. Let r be a rank array with the above dimension vector. Then for all compositions β ∈ M(v(r)),

D_{Hom} ⊂ β_{+} ⊂ D(Ω_{0}) and (7.12)

(7.13)  β = w_{0}β_{+},

where w_{0} is the block long permutation, reversing the row indices within each block row.

Proof. Let β ∈ M(v(r)). Equation (7.13) is a consequence of Lemma 7.5 and Proposition 1.6, as Zelevinsky permutations have no descents in each block row.

Both of the partition diagrams D_{r} ↑← and D_{r} ←↑ are contained in D(Ω_{0}), by Definition 1.10 along with (1.3), (1.4), (1.7), and (1.8). Proposition 7.4 implies that β_{+} ⊂ D(Ω_{0}).

Since D_{Hom} ⊂ D_{r}, the monomial x^{D_{Hom}} divides 𝔖_{v(r)}. By Lemma 7.6, D_{Hom} ≤ β. But the partition D_{Hom} is constant in each block row. With (7.13), it follows that D_{Hom} ⊂ β_{+}. □

Finally, we observe that each Demazure character appearing in 𝔖_{v(r)} is the monomial x^{D_{Hom}} times a product of Schur polynomials.

Definition 7.8. Assume that λ = β_{+} for some composition β ∈ M(v(r)). The skew shape λ/D_{Hom} afforded by (7.12) consists of a union of partition diagrams in which the partition λ_{i} is contained in the r_{i−1} × r_{i} rectangle comprising the i^{th} block row of D(Ω_{0})/D_{Hom}. In this situation, we say that the partition list \underline{λ} = (λ_{1}, λ_{2}, …, λ_{n}) is obtained by deleting D_{Hom} from λ, and we write \underline{λ} = λ − D_{Hom}.

Proposition 7.9. Suppose that β is a composition in M(v(r)), that λ = β_{+}, and that \underline{λ} is obtained from λ by deleting D_{Hom}. Then

κ_{β} = x^{D_{Hom}} ∏_{i=1}^{n} s_{λ_{i}}(x^{i−1}).

Proof. Break D_{Hom} into a sequence (μ_{0}, …, μ_{n}) of shapes, where μ_{i} ∈ ℕ^{r_{i}} is rectangular (all of its parts are equal) and μ_{n} = (0^{r_{n}}). Let w_{0}^{i} be the longest element of the symmetric group S_{r_{i}} acting on the i^{th} block of row indices. By (7.1) and (7.4) we have

κ_{β} = π_{w_{0}}x^{λ}
= ∏_{i=0}^{n−1} π_{w_{0}^{i}}(x^{i})^{λ_{i+1}+μ_{i}}

= ∏_{i=0}^{n−1}(x^{i})^{μ_{i}}s_{λ_{i+1}}(x^{i})
= x^{D_{Hom}} ∏_{i=1}^{n} s_{λ_{i}}(x^{i−1}).  □

7.3. Quiver constants are Stanley coefficients

Having an expansion of the double quiver function as a sum of products of Stanley symmetric functions automatically produces an expansion as a sum of products of Schur functions. Next, in Theorem 7.10, we shall see that the coefficients in this expansion are the Buch–Fulton quiver constants. Using this result, we show in Theorem 7.14 that quiver constants are special cases of Stanley coefficients.

For a list \underline{λ} = (λ_{1}, λ_{2}, …, λ_{n}) of partitions and two sequences x = (x^{0}, …, x^{n}) and ∘y = (y^{n}, …, y^{0}) of infinite alphabets, let

s_{\underline{λ}}(x − ^{∘}y) = ∏_{i=1}^{n} s_{λ_{i}}(x^{i−1} − y^{i})

be the product of Schur functions in differences of alphabets. This notation parallels that with y = x in (1.16), and for products of Stanley symmetric functions in Section 6.1. We follow the conventions after (6.2) for finite alphabets.

Theorem 7.10. If c_{\underline{λ}}(r) is the quiver constant from Theorem 1.20, then

F_{r}(x − ^{∘}y) = ∑_{\underline{λ}} c_{\underline{λ}}(r)s_{\underline{λ}}(x − ^{∘}y).

Moreover, using the Stanley coefficients α_{w}^{λ} from (6.1) and writing α\underline{λ} ; w = ∏_{i=1}^{n} α_{w_{i}}^{λ_{i}}, we get

(7.14)  c_{\underline{λ}}(r) = ∑_{w∈W (r)} α\underline{λ} ; w.

Proof. Expanding the right-hand side of Theorem 6.20 into Schur functions yields

F_{r}(x − ^{∘}y) = ∑_{w∈W (r)} ∑_{\underline{λ}} α\underline{λ} ; ws_{\underline{λ}}(x − ^{∘}y).

Specializing ^{∘}y to ^{∘}x yields an expression for F_{r}(x − ^{∘}x) as a sum of Schur function products s_{\underline{λ}}(x −^{∘}x). By Corollary 6.23 and Theorem 1.20, uniqueness in the Main Theorem of [BF99] implies that the coefficient ∑_{w∈W (r)} ∑_{\underline{λ}} α\underline{λ} ; w on s_{\underline{λ}}(x − ^{∘}x) in this expression is c_{\underline{λ}}(r). □

Remark 7.11. Theorem 7.10 and Corollary 6.23 are the only results in this paper that logically depend on the Main Theorem of [BF99], other than Theorem 1.20 (which needs the Main Theorem of [BF99] for its statement). In other words, starting from the multidegree characterization of quiver polynomials in Definition 1.18, the statements and proofs of all of our other combinatorial formulae—including all double and stable versions, as well as our combinatorial formula for the quiver constants to come in Theorem 7.21—are independent from [BF99]. It is only to identify the constants on the right side of (7.14) as the quiver constants appearing in the Conjecture from [BF99] that we apply Theorem 1.20, and hence the Main Theorem of [BF99].

Remark 7.12. The quiver constants c_{\underline{λ}}(r) may be computed fairly efficiently using (7.14).

We need a proposition, in which our choice of notation (using x−^{∘}x in arguments of quiver polynomials) should finally become clear: we use Q_{r}(x) and F_{r}(x) to denote the y = 0 specializations (as opposed to y = x specializations) of the double quiver polynomial Q_{r}(x − ^{∘}y) and the double quiver function F_{r}(x − ^{∘}y). The “doubleness” of these expressions is crucial here, because in the ordinary single case it is impossible in the summation formulae to set just the second set of variables in each factor of every summand to zero.

Proposition 7.13. If r is any rank array, then Q_{r}(x) = F_{r}(x_{r}) is the finite x alphabet specialization of the double quiver function at y = 0.

Proof. Calculate Q_{m+r}(x) and Q_{r}(x) using Theorem 5.5. Consider a reduced pipe dream D ∈ RP(v(m+r)) that happens to lie entirely in rows indexed by x_{r} (as opposed to x_{m+r}). Restricting attention to the (j − 1)^{st} block row of D for the moment, let w_{j} satisfy w_{j}(D) = m + w_{j} using Corollary 5.16. No cross can be farther to the left of the vertical dividing line in that block row (from after Lemma 5.11) than the leftmost in the upside-down (rotated by 180^{∘}) top pipe dream for w_{j}. Therefore, the miniature pipe dream in block row j−1 of D has at least m blank columns on its left. By Proposition 5.15, horizontally shifting all crosses m cells to the left therefore induces a canonical bijection onto RP(v(r) from the set of pipe dreams in RP(v(m + r)) that happen to lie entirely in rows indexed by x_{r}. Corresponding pipe dreams obviously yield the same monomial in x. Therefore setting all variables in the set x_{m+r} ∖x_{r} to zero in Q_{m+r}(x) yields Q_{r}(x). Now use Corollary 6.13. □

Theorem 7.14. The quiver constant c_{\underline{λ}}(r) is the Stanley coefficient α_{v(r)}^{λ}, where λ is the unique partition such that the partition list \underline{λ} is obtained from λ by deleting D_{Hom}.

Proof. Multiplying through by the implicit denominator in Proposition 7.13 yields

(7.15)  𝔖_{v(r)}(x) = 𝔖_{v(Hom)}(x) ∑_{\underline{λ}} c_{\underline{λ}}(r)s_{\underline{λ}}(x_{r}),

where c_{\underline{λ}}(r) is the quiver constant from Theorem 1.20 by Theorem 7.10. Proposition 7.9 shows that the Demazure character expansion of 𝔖_{v(r)}(x) has the same form as the right side of (7.15). Linear independence of the polynomials s_{\underline{λ}}(x_{r}) therefore implies that (7.15) is the Demazure character expansion. Equate coefficients on each Demazure character in Theorem 7.2 and (7.15) and apply Corollary 7.3. □

Remark 7.15. The comparison of (7.15) with Theorem 7.2 via Proposition 7.9 occurs at the level of ordinary Schubert polynomials (in a single set x of variables) rather than double Schubert polynomials (in x and y). This restriction is forced upon us by the fact that the Demazure character theory behind Proposition 7.9 has not been sufficiently developed in the double setting. (See [Las03] for the beginnings of such a ‘double’ theory.) Since the comparison of (7.15) with Theorem 7.2 is the key point in the proof of Theorem 7.14, it is our single most important motivation for developing the double version of quiver polynomials via the ratio formula: we must be able to set y = 0 for this comparison.

7.4. Peelable tableaux

Peelable tableaux [RS95a, RS98] arose from Stanley’s problem of counting reduced decompositions of permutations [Sta84], the expansion by Lascoux and Schützenberger of the Schubert polynomial as a sum of Demazure characters [LS89], and from Magyar’s character formula for the global sections of line bundles over Bott–Samelson varieties [Mag98]. Here we use them to give our first combinatorial formula for quiver constants. Although other combinatorial interpretations of Stanley coefficients would also produce formulae, it is the peelable tableaux that will connect to factor sequences in Section 8.

Let D be a diagram, meaning a finite set of pairs of positive integers. The diagram D is northwest if (i_{1}, j_{2}) ∈ D and (i_{2}, j_{1}) ∈ D implies (i_{1}, j_{1}) ∈ D for i_{1} < i_{2} and j_{1} < j_{2}; thus, if two cells lie in D, then so does the northwest corner of the smallest rectangle containing both. Permutation diagrams are always northwest. The diagram D(λ) of a partition λ consists of λ_{i} left-justified cells in the i^{th} row for each i.

For tableaux Q and P, write Q ⊃ P if Q contains P in its northwest corner. Write Q−P for the skew tableau obtained by removing the subtableau P from Q.

A column of a diagram can be identified with the set of indices of rows in which it has a cell. This subset can also be identified with the decreasing word consisting of these row indices. As in Section 7.1, denote by [u] the unique tableau Knuth-equivalent to a word u.

Now consider a northwest diagram D. A tableau Q is D-peelable provided that:

1 1 1 1 1 1 1 5 2 2 2 2 2 2 2 3 3 3 5 5 4 4 4 5 5 5 8 9 9 9 ⟶  1 1 1 1 1 1 5 2 2 2 2 2 2 3 3 5 5 4 4 9 5 5 8 9 9 ⟶  1 1 1 1 1 5 2 2 2 2 2 3 5 5 4 9 9 5 8 9 ⟶

1 1 1 1 5 2 2 2 2 5 5 9 9  ⟶  1 1 1 5 2 2 2 5 5 9 9  ⟶  1 1 5 2 2 5 5 9 9  ⟶

1 5 2 9 5 9  ⟶  5 9  ⟶ ∅

Figure 4. Peeling a peelable tableau

1. Q is the empty tableau when D is the empty diagram; or else

2. Q ⊃ C and [Q − C] is (D − C)-peelable when C is the first nonempty column of D. We refer to the map Q ↦ [Q − C] as peeling, and denote by Peel(D) the set of D-peelable tableaux of partition shape.

Remark 7.16. The weight of any D-peelable tableau is the code of D, meaning the sequence code(D) = (c_{1}, c_{2}, …) in which c_{i} is the number of cells in row i of D.

Example 7.17. The tableau in Fig. 4 is D_{r}-peelable as exhibited, where r is the rank array in Example 1.5, and the diagram D_{r} is in Example 1.9. The sequences of removed columns C (appearing in boldface below) are given by the columns of D_{r}, after each cell has been labeled by its row index (this labeling of cells in D_{r} is depicted explicitly in Example 8.1, below).

Remark 7.18. The set of D-peelable tableaux may be constructed by “unpeeling” as follows. Let C be as in the definition of D-peelability, and p the number of cells in C. Suppose all of the (D − C)-peelable tableaux have been constructed. For each pair (T, V) where T is a (D − C)-peelable tableau and V is a vertical p-strip (a skew shape p cells, no two of which lie in the same row) whose union with the shape of T is a partition, use the jeu de taquin [Ful97, Section 1.2] to slide T into the topmost cell of V, then into the next topmost cell of V, and so on. This vacates p cells at the top of the first column. Place the single-column tableau C into the vacated cells. If the result is a (semistandard) tableau then it is D-peelable by construction. All D-peelable tableaux can be obtained in this manner.

Theorem 7.19. [RS95a, Theorem 29] For any (partial) permutation w,

𝔖_{w}(x) = ∑_{Q∈Peel(D(w))} κ_{wt(K_{−}(Q))}.

Remark 7.20. Theorem 7.19 gives a combinatorial description of the multiset M(w) in Theorem 7.2: it is the multiset of compositions given by wt(K_{−}(Q)) for Q ∈ Peel(D(w)).

Theorem 7.21. The quiver constant c_{\underline{λ}}(r) is the number of D_{r}-peelable tableaux of shape λ, where \underline{λ} is obtained from λ by deleting D_{Hom}.

Proof. This follows from Theorems 7.14 and 7.19 along with Corollary 7.3, since for any tableau Q, the partition wt(K_{−}(Q))_{+} is just the shape of Q. □

Section 8. Factor sequences from peelable tableaux

In [BF99] it was conjectured that the quiver constant c_{\underline{λ}}(r) counts certain lists of tableaux called ‘r-factor sequences of shape \underline{λ}’ (see the end of Section 8.1 for a precise statement). We already know by Theorem 7.21 that the quiver constants count peelable tableaux. We shall verify a variant (Corollary 8.23) of the Buch–Fulton conjecture by establishing a bijection from peelable tableaux to factor sequences.

8.1. Factor sequences

The tableau array T (r) = (T_{ij}(r)) for the rank array r is defined as follows. View the diagram D_{r} = D(v(r)) of the Zelevinsky permutation as being filled with integers, where every cell (either ∗ or □) in row k is filled with k ∈ ℤ. Define T_{ij}(r) to be the rectangular tableau given by the i^{th} block column (from the right) and the j^{th} block row (from the top), defined for blocks on or below the superantidiagonal. The tableau T_{ij} has shape R_{i−1,j+1} by Lemma 1.8.

For later use, we define tableaux Y_{i} and K_{i} as follows. Let Y be the tableau given by restricting the filling of D_{r} to the blocks above the superantidiagonal, and set Y_{i} equal to the part of Y in the i^{th} block column. Let K_{i} be the tableau obtained by stacking (leftjustified) the tableaux in the i^{th} column of the tableau array, with T_{i,i−1} on top, T_{i,i} below it, and so on. Equivalently,

(8.1)  K_{i} = [T_{i,n−1}T_{i,n−2} ⋯ T_{i,i}T_{i,i−1}].

Let A_{i} be the interval of row indices occurring in the i^{th} block row, and then set B_{i} = A_{i} ∪ A_{i+1} ∪ · · ·. Given a set A, let A. denote the set of words in the alphabet A. Then

(8.2)  Y_{i} ∈ (A_{0} ∪ ⋯ ∪ A_{i−2})..

Example 8.1. For the ranks r in Example 1.5, with diagram D_{r} in Example 1.9, Fig. 5 depicts the filling of D_{r}, the tableau array T = T (r), and the tableaux Y_{i} and K_{i}. □

We now recall the recursive structure underlying the definition of an r-factor sequence. Let r̂ be the rank array obtained by removing the entries r_{ii} for 0 ≤ i ≤ n. Using notation from Definition 1.10, observe that D□ ; r̂ is obtained from D_{r}^{□} by removing the superantidiagonal cells and some empty rows and columns. Hence we may identify D□ ; r̂ with a subdiagram of D_{r}^{□} by reindexing the nonempty rows. Under this identification the tableau array T (r̂) is obtained from T (r) by removing the superantidiagonal tableaux.

Example 8.2. Continuing with Example 8.1, we get r̂, R̂, Dr̂, and T (r̂) as in Fig. 6. □

Returning to the general case, let T_{i} = T_{i,i−1} for 1 ≤ i ≤ n.

Definition 8.3. The notion of r-factor sequence is defined recursively, as follows. 1. If n = 1 then there is a unique r-factor sequence, namely (T_{1}).

filled D_{r} = \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & × & · & · & · & · \\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & · & × & · & · & · \\ 3 & 3 & 3 & × & · & · & · & · & · & · & · & · \\ 4 & 4 & 4 & · & × & · & · & · & · & · & · & · \\ 5 & 5 & 5 & · & · & 5 & 5 & · & · & 5 & × & · \\ × & · & · & · & · & · & · & · & · & · & · & · \\ · & × & · & · & · & · & · & · & · & · & · & · \\ · & · & 8 & · & · & × & · & · & · & · & · & · \\ · & · & 9 & · & · & · & 9 & · & · & 9 & · & × \\ · & · & × & · & · & · & · & · & · & · & · & · \\ · & · & · & · & · & · & × & · & · & · & · & · \\ · & · & · & · & · & · & · & · & · & × & · & · \end{matrix} T = \begin{aligned}3 & 2 & 1 & i⧸j \\ & & ∅ & 0 \\ & 5 5 & 5 & 1 \\ 8 9 & 9 & 9 & 2 \end{aligned}

\begin{aligned}A_{2} & = & \{6, 7, 8, 9\} & A_{1} & = & \{3, 4, 5\} & A_{0} & = & \{1, 2\} \\ Y_{3} & = & 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 & Y_{2} & = & 1 1 1 1 2 2 2 2 & Y_{1} & = & ∅ \\ K_{3} & = & 8 9 & K_{2} & = & 5 5 9 & K_{1} & = & 5 9 \end{aligned}

Figure 5. Tableau array and other data from Example 8.1

r̂ = \begin{aligned}& & & · \\ & & · & 2 \\ & · & 2 & 1 \\ · & 2 & 1 & 0 \end{aligned}  R̂ = \begin{aligned}& & · \\ & · \\ · \end{aligned}

Dr̂ = \begin{matrix}∗ & ∗ & · & · & · & · \\ ∗ & ∗ & · & 5 & · & · \\ · & · & · & · & · & · \\ · & 9 & · & 9 & · & · \\ · & · & · & · & · & · \\ · & · & · & · & · & · \end{matrix}  T (r̂) = \begin{aligned}& & · \\ & · & 5 \\ · & 9 & 9 \end{aligned}

Figure 6. Data obtained by deleting the antidiagonal of r from Fig. 5

2. For n ≥ 2, an r-factor sequence is a tableau list (W_{1}, …, W_{n}) such that there exists an r̂-factor sequence (U_{1}, …, U_{n−1}) (for the tableau array T (r̂) as defined above) and factorizations

(8.3)  U_{i} = [P_{i}Q_{i}]  for 1 ≤ i ≤ n − 1

such that

(8.4)  W_{i} = [Q_{i}T_{i}P_{i−1}]  for 1 ≤ i ≤ n,

\begin{aligned}& & ∅ \\ & 5 5 & 5 \\ 8 9 & 9 & 9 \end{aligned} ⟶^{factor} \begin{aligned}& & ∅ \\ & 5 5 & 5 \\ 8 9 & 9 & 9 ·∅ \end{aligned} ⟶^{move} \begin{aligned}& & ∅ \\ & 5 5 & ∅· 5 \\ 8 9 & 9 · 9 \end{aligned} ⟶^{mult} \begin{aligned}& & ∅ \\ & 5 5 & 5 \\ 8 9 & 9 9 \end{aligned}

⟶^{factor} \begin{aligned}& & ∅ \\ & 5 5 & ∅· 5 \\ 8 9 & 9 9 ·∅ \end{aligned} ⟶^{move} \begin{aligned}& & 5 ·∅ \\ & ∅· 5 5 ·∅ \\ 8 9 · 9 9 \end{aligned} ⟶^{mult} 5 5 5 8 9 9 9

Figure 7. Computation of a factor sequence

where by convention

(8.5)  Q_{n} = P_{0} = ∅.

The global computation of a factor sequence proceeds as follows. Index the block antidiagonals by their distance (going northwest) to the main superantidiagonal.

1. Initialize W_{ij} := T_{ij} for all n ≥ j + 1 ≥ i ≥ 1.

2. Set k := n − 1.

3. At this point the k^{th} antidiagonal of W is an r^{(k)}-factor sequence, where r^{(k)} is obtained from r by removing the 0^{th} through (k − 1)^{st} antidiagonals. Factor each tableau W_{ij} = [P_{ij}Q_{ij}] on the k^{th} antidiagonal. Now move the left factor P_{ij} to the west and the right factor Q_{ij} to the north.

4. At this point W_{ij} = T_{ij} on the (k−1)^{st} antidiagonal. For every position on the (k−1)^{st} antidiagonal, multiply the current entry T_{ij} on the left with the tableau Q_{i,j+1} coming from the south and on the right with the tableau P_{i−1,j} coming from the east. That is, set W_{ij} := [Q_{i,j+1}T_{ij}P_{i−1,j}].

5. Set k := k − 1. If k = 0, stop. Otherwise go to step 3.

Example 8.4. In Fig. 7 we compute a factor sequence as above, except that after a tableau is factored and its factors moved, its position is vacated. Compare Example 8.14. □

Remark 8.5. It is clear from the global computation that for any r-factor sequence (W_{1}, …, W_{n}), the letters of W_{i} are a submultiset of the letters from the tableaux T_{kℓ} to the south and east of the i^{th} block column and (i − 1)^{st} block row (that is, for ℓ + 1 ≥ i ≥ k).

We now recall the original definition of factor sequence and the associated conjecture in [BF99]. The tableau we index by T_{ij} is indexed as T_{n−1−j,n+1−i} in [BF99].

Let (T_{ij}) be any array of semistandard tableaux with T_{ij} of shape R_{i−1,j+1} (which in turn is defined in terms of r in Section 1.2). Say that a (T_{ij})-factor sequence is any sequence of tableaux obtained from the factor-move-multiply algorithm described above, but starting from the array (T_{ij}) rather than our canonical array (T_{ij}(r)) of semistandard tableaux. Say that (T_{ij}) satisfies the factor sequence condition if the number of (T_{ij})-factor sequences of shape \underline{λ} is equal to the quiver constant c_{\underline{λ}}(r). The Buch–Fulton factor sequence conjecture asserts that the factor sequence condition holds for tableau arrays (T_{ij}) such that the tableaux collectively have no repeated entries and that each entry of T_{ij} is larger than each entry of T_{i,j−1} and of T_{i+1,j}.

Corollary 8.23 states that the factor sequence condition holds for the canonical semistandard tableau array (T_{ij}(r)). This result is not a special case of the Buch–Fulton conjecture, nor does the former immediately imply the latter, although it does rather easily imply that the factor sequence condition holds for a number of tableau arrays encompassed by the Buch–Fulton conjecture. It is an interesting and purely combinatorial problem to find the largest class of tableau arrays (T_{ij}) which satisfy the factor sequence condition.

8.2. Zelevinsky peelable tableaux

This section contains some scattered but essential results regarding peelable tableaux. Our discussion culminates in Definition 8.13, which says how to construct sequences of tableaux from peelable tableaux associated to the diagram of a Zelevinsky permutation.

The following lemma generalizes the dominance bounds for D(w)-peelable tableaux implied by Proposition 7.4 to the case of peelables with respect to any northwest diagram. This added generality will be used in the proof of Proposition 8.17. Recall the definitions of ⊴, D ↑, and D ← given just before Proposition 7.4.

Lemma 8.6. If a D-peelable tableau for a northwest diagram D has shape λ, then

D ←↑ ⊴ λ ⊴ D ↑← .

Proof. If a tableau has shape λ and weight γ, then γ_{+} ⊴ λ [Ful97, Section 2.2]. Noting that D ←↑ = code(D)_{+}, the first dominance condition holds by Remark 7.16. The second condition is a consequence of the first, using the following facts.

1. The transposing of shapes gives an anti-isomorphism of the poset of partitions under the dominance partial order.

2. There is a shape-transposing bijection Peel(D) → Peel(D^{t}), where D^{t} is the transpose diagram of D [RS98, Definition–Proposition 31]. □

There are a number of operations D → D′ one may perform on diagrams that induce maps Peel(D) → Peel(D′) on the corresponding sets of peelable tableaux. For notation, let s_{r}D be the diagram obtained by exchanging the r^{th} and (r + 1)^{st} rows of D, and similarly for Ds_{r} exchanging columns. Given a diagram D and a set I of row indices, let D|_{I} be the diagram obtained by taking only the rows of D indexed by I. If u is a word or a tableau, let u|_{I} be the word obtained from u by erasing all letters not in I, and set [u]_{I} = [u|_{I}].

Proposition 8.7. Let D be a northwest diagram.

1. [RS98, Definition–Proposition 13] If s_{r}D is also northwest, then there is a shapepreserving bijection s_{r} : Peel(D) → Peel(s_{r}D). (It is given by the action of the “r^{th} type A crystal reflection operator”; see [RS98, Section 2] for the definition of s_{r}.)

2. [RS98, Lemma 51] If both D and Ds_{r} are northwest then Peel(Ds_{r}) = Peel(D).

3. [RS98, Lemmata 54 and 56] There is a surjective map Peel(D) → Peel(D|_{I}) given by Q ↦ [Q]_{I}.

Lemma 8.8. Let D be a northwest diagram such that D ⊃ D(μ) for a partition μ. Then Q ⊃ key(μ) for all tableaux Q ∈ Peel(D).

Proof. Suppose μ has i nonempty rows. By Proposition 8.7.3 with I = {1, 2, …, i}, we may assume that D has i rows. Since D ⊃ D(μ), the first column C of D is C = i ⋯ 321. Let ν = (μ_{1} − 1, μ_{2} − 1, …, μ_{i} − 1). The first column of Q contains C and contains only letters in I, so it must be equal to C. Then [Q − C] is merely Q with its first column erased. By definition [Q − C] is (D − C)-peelable. By induction Q − C ⊃ key(ν). But then Q ⊃ C key(ν) = key(μ). □

We say that a diagram D is λ-partitionlike for the partition λ if D is northwest and D(λ) can be obtained from D by repeatedly exchanging adjacent rows or columns, staying within the family of northwest shapes.

Lemma 8.9. Peel(D) consists of one tableau, namely key(code(D)), if D is partitionlike.

Proof. Using parts 1 and 2 of Proposition 8.7 to pass from D to D(λ), there is a shapepreserving bijection from Peel(D) to Peel(D(λ)). Lemma 8.8 and Remark 7.16 imply that key(λ) is the unique D(λ)-peelable tableau. Thus Peel(D) consists of a single tableau of shape λ (since key(λ) has shape λ) and weight code(D) (by Remark 7.16). But key(code(D)) is the unique such tableau. □

Lemma 8.10. Suppose D is a northwest diagram such that its r^{th} row (viewed as a subset of integers given by the column indices of its cells) either contains or is contained in its (r + 1)^{st} row. Suppose that rows r and r + 1 have a and b cells, respectively. If Q is a D-peelable tableau, then [Q]_{{r,r+1}} = [(r + 1)^{b}r^{a}].

Proof. Proposition 8.7.3 with I = {r, r + 1} implies [Q]_{I} is D|_{I}-peelable. By Lemma 8.9 there is a unique D|_{I}-peelable tableau, namely [(r + 1)^{b}r^{a}]. □

In light of Remark 7.20, the following two results are restatements of Lemma 7.5 and Proposition 7.7.

Lemma 8.11. Let w be a permutation such that w(i) < w(i + 1). If Q ∈ Peel(D(w)) and β = wt(K_{−}(Q)), then β_{i} ≤ β_{i+1}.

Proposition 8.12. For a fixed dimension vector (r_{0}, r_{1}, …, r_{n}), identify D_{Hom} and D(Ω_{0}) with partitions of those shapes. Let r be a rank array with the above dimension vector, Q ∈ Peel(D_{r}), and λ = shape(Q). Then D_{Hom} ⊂ λ ⊂ D(Ω_{0}).

Let Q ∈ Peel(D_{r}) and λ = shape(Q). By Lemma 8.8 Q ⊃ key(D_{Hom}). The skew tableau Q − key(Hom) has shape λ/D_{Hom} ⊂ D(Ω_{0})/D_{Hom}. The skew shape D(Ω_{0})/D_{Hom} is a disjoint union of r_{i−1} × r_{i} rectangles for 1 ≤ i ≤ n in which the r_{i−1} × r_{i} rectangle resides in the (i − 1)^{st} block row and i^{th} block column, where the indexing of block rows and columns is the same as in Section 1.2. Proposition 8.12 and Lemma 8.8 imply that the following map Ψ_{r} from Peel(D_{r}) to length n sequences of tableaux is well-defined. It is the peelable tableau analogue of Definition 7.8.

Definition 8.13. Let Q be a D_{r}-peelable tableau. The sequence Ψ_{r}(Q) = (W_{1}, W_{2}, …, W_{n}) of tableaux is obtained by deleting D_{Hom} from Q if W_{i} is the subtableau of the skew tableau Q − key(D_{Hom}) sitting inside the r_{i−1} × r_{i} rectangle.

Example 8.14. The image under Ψ_{r} of the D_{r}-peelable tableau in Example 7.17 is the sequence (W_{1}, W_{2}, W_{3}) depicted in Fig. 8. □

8.3. Bijection to factor sequences

Next we give a condition for a tableau list to be obtained by deleting D_{Hom} from a D_{r}- peelable tableau. This will be indispensable for proving the bijection from peelable tableaux to factor sequences, the aim of this section. The idea is to perform several peelings together, so as to remove one block column at a time. The ‘code of D’ is defined in Remark 7.16.

Lemma 8.15. Let D be a northwest diagram, divided by a vertical line into diagrams D^{w} (“D-west”) and D^{e} (“D-east”), where D^{w} is the first several columns of D. Assume D^{w} is λ-partitionlike, and set Y = key(code(D^{w})).

W_{3} = 8 9 9 9  W_{2} = 5 5  W_{1} = 5

1 1 1 1 1 1 1 5 2 2 2 2 2 2 2 3 3 3 5 5 4 4 4 5 5 5 8 9 9 9 ⟼ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 5 ; ∗ ∗ ∗ ∗ ∗ ∗ ∗ ; ∗ ∗ ∗ 5 5 ; ∗ ∗ ∗ ; ∗ ∗ ∗ ; 8 9 9 ; 9

Figure 8. Peelable tableau to factor sequence under Ψ_{r}

1. If Q ∈ Peel(D) then Q ⊃ Y.

2. If Q ⊃ Y, then Q ∈ Peel(D) if and only if [Q − Y] ∈ Peel(D^{e}).

Proof. By Proposition 8.7.2 we assume that the columns of D^{w} are decreasing with respect to containment. Suppose the rows of D^{w} are not decreasing with respect to containment. Then some row r of D^{w} is a proper subset of row r + 1. Since D is northwest, row r of D^{e} is empty. This given, apply Proposition 8.7.1. It is straightforward to check that the above conditions for D and Q are equivalent to those for s_{r}D and s_{r}Q. Hence we may assume that D^{w} = D(λ) for a partition λ.

Now item 1 follows from Lemma 8.8. For item 2 we use the jeu de taquin [Ful97, Section 1.2]. Let C_{j} be the column word that gives the (decreasing) set of row indices for the elements in column j of D. The result [Q − C_{1}] of one peeling step can be computed by sliding the skew tableau Q−C_{1} into the cells occupied by C_{1}. Repeating this process for the columns of D^{w}, the tableau resulting from applying the corresponding peeling steps to Q is also obtained by removing the subtableau Y (which must be present in Q by item 1) and sliding the skew tableau Q − Y into the cells that were occupied by Y. □

Lemma 8.16. If (W_{1}, …, W_{n}) = Ψ_{r}(Q) for Q ∈ Peel(D_{r}), then

(8.6)  W_{i} ∈ B._{i−1}
(8.7)  height(W_{i}) ≤ r_{i−1}
(8.8)  width(W_{i}) ≤ r_{i},

and Q has the block column factorization

(8.9)  Q = (W_{n}Y_{n}) ⋯ (W_{2}Y_{2})(W_{1}Y_{1}).

Proof. Statement (8.6) follows from (8.2) and the fact that W_{i} sits below Y_{i} in a block column of the tableau Q. The rest follows by definition, using Proposition 8.12. □

Proposition 8.17. The list (W_{1}, …, W_{n}) of tableaux is obtained from some D_{r}-peelable tableau by deleting D_{Hom} if and only if there exist tableaux X_{1}, …, X_{n} such that

(8.10)  X_{n} = W_{n},
(8.11)  X_{i} ⊃ K_{i} for all i,
(8.12)  X_{i} = [(X_{i+1} − K_{i+1})W_{i}] for i = 1, …, n − 1,
(8.13)  and X_{1} = K_{1};

and if

(8.14)  Z_{i} = [X_{i}W_{i−1} ⋯ W_{2}W_{1}]_{B_{i−1}}

then

(8.15)  height(Z_{i}) ≤ r_{i−1},
(8.16)  width(Z_{i}) ≤ r_{i},
(8.17)  and wt(Z_{i}) = wt(T_{≤i, ≥i−1}),

where wt(T_{≤i, ≥j}) = ∑_{\begin{matrix}k≥i \\ ℓ≤j \end{matrix}} wt(T_{k,ℓ}).

The proof will come shortly.

Lemma 8.18. The conditions in Proposition 8.17 imply the following:

(8.18)  X_{i} ∈ B._{i−1}
(8.19)  height(X_{i}) ≤ r_{i−1}
(8.20)  width(X_{i}) ≤ r_{i}
(8.21)  wt(X_{i}) ≤ wt(T_{≤i,≥i−1})

Proof. (8.18) can be proved by induction using (8.6). X_{i} is a factor of Z_{i} by (8.18) and (8.14). The last three assertions then follow from (8.15), (8.16), and (8.17). □

Lemma 8.19. If (8.10) through (8.13) hold, then the conditions (8.18) through (8.20) imply (8.6) through (8.9), plus

wt(W_{i}) ≤ wt(T_{≤i, ≥i−1}).

This last inequality is the same condition on (W_{1}, …, W_{n}) appearing in Remark 8.5.

Example 8.20. The tableaux X_{i} and Z_{i} associated with the above tableaux W_{i} are given by

(8.22)  X_{3} = 8 9 9 9  X_{2} = 5 5 9 9  X_{1} = 5 9

(8.23)  Z_{3} = 8 9 9 9  Z_{2} = 5 5 5 9 9  Z_{1} = 5 9

Note that the X tableaux appear at the bottom of the leftmost block columns in the first, fourth, and eighth tableaux in Example 7.17. □

Proof of Proposition 8.17. Set D = D_{r}, and let D_{i} be the diagram obtained from D by removing the n^{th} through (i+1)^{st} block columns (or equivalently the leftmost r_{n} +⋯+r_{i+1} columns). Let D_{i,j} be obtained from D_{i} by removing the 0^{th} through (j − 1)^{st} block rows. In other words, D_{i,j} consists of the block at the intersection of the i^{th} block column and j^{th} block row, along with all blocks to its south and east.

Suppose (W_{1}, …, W_{n}) = Ψ_{r}(Q) for a tableau Q ∈ Peel(D). Let Q_{i} ∈ Peel(D_{i}) be the tableau obtained from Q by r_{n} + ⋯ + r_{i+1} peelings. We shall proceed by induction, with hypothesis (for descending i) being that Q_{j} has the block column factorization

(8.24)  Q_{j} = (X_{j}Y_{j})(W_{j−1}Y_{j−1}) ⋯ (W_{2}Y_{2})(W_{1}Y_{1})

and that X_{j}, Z_{j} have been defined and have the desired properties for n ≥ j ≥ i.

Suppose 0 ≤ i ≤ n and the induction hypothesis holds for indices greater than i. It must be shown that the induction hypothesis holds for i. The tableau X_{i} is defined by (8.10) when i = n, or by (8.12) when i < n (note that (8.11) holds for i + 1).

Apply Lemma 8.15 to Q_{i+1} ∈ Peel(D_{i+1}), with D_{i+1}^{w} the west block column (the leftmost r_{i+1} columns) of D_{i+1}. By (8.24) for i + 1, the first block column of Q_{i+1} is X_{i+1}Y_{i+1}. Also key(code(D_{i+1}^{w})) = K_{i+1}Y_{i+1}. Recalling that Q_{i} is defined by iterated peelings from Q, we have

(8.25)  \begin{aligned}Q_{i} & = & [(X_{i+1}Y_{i+1} − K_{i+1}Y_{i+1})(W_{i}Y_{i}) ⋯ (W_{1}Y_{1})] \\ & = & [(X_{i+1} − K_{i+1})W_{i}Y_{i} ⋯ (W_{1}Y_{1})] \\ & = & [(X_{i}Y_{i})(W_{i−1}Y_{i−1}) ⋯ (W_{1}Y_{1})], \end{aligned}

the last equality by definition of X_{i}. Apply Proposition 8.7.3 to Q_{i} ∈ Peel(D_{i}) and the interval B_{i−1}. Note that (D_{i})|_{B_{j}} = D_{i,j}, so in particular [Q_{i}]_{B_{i−1}} ∈ Peel(D_{i,i−1}). Thus

[Q_{i}]_{B_{i−1}} = [X_{i}W_{i−1} ⋯ W_{1}]_{B_{i−1}} = Z_{i},

by (8.25), (8.2), and (8.14). We have proved that

(8.26)  Q ∈ Peel(D) ⇒ Z_{i} ∈ Peel(D_{i,i−1}).

The sum of the heights of the rectangles in the i^{th} block column of D_{i,i−1} ⊂ D is at most r_{i−1} by (1.3). The sum of the widths of the rectangles in the (i − 1)^{st} block row of D_{i,i−1} ⊂ D is at most r_{i} by (1.4). Z_{i} fits inside a r_{i−1} × r_{i} rectangle, by Lemma 8.6, (1.7), and (1.8). This proves (8.15) and (8.16). Equation (8.17) follows from Remark 7.16. By Lemma 8.18, the corresponding properties (8.18) through (8.21) hold for the X tableaux. The block column factorization (8.24) for j = i then follows from (8.25), (8.19), (8.20), and (8.18).

To prove (8.11), apply Lemma 8.15 with Q_{i} ∈ Peel(D_{i}) and D_{i}^{w} given by the westmost block column of D_{i}. This yields that Q_{i} ⊃ key(code(D_{i}^{w})) = K_{i}Y_{i}. By (8.24) the first block column of Q_{i} is X_{i}Y_{i}, from which (8.11) follows.

For the converse (i.e. that (8.10)–(8.17) imply (W_{1}, …, W_{n}) lies in the image of Ψ_{r}), suppose the X_{i} and Z_{i} exist and have the desired properties. By Lemmas 8.18 and 8.19, conditions (8.18)–(8.21) and (8.6)–(8.8) all hold. Let Q_{i} be defined by the block column factorization (8.24). The aforementioned properties guarantee that Q_{i} is a tableau. It will be shown by induction on increasing i that Q_{i} ∈ Peel(D_{i}).

For i = 0, Q_{0} is empty by (8.24), and D_{0} is the empty diagram so Q_{0} ∈ Peel(D_{0}). Suppose i > 0. Apply Lemma 8.15 to D_{i} and Q_{i} with D_{i}^{w} the west block column of D_{i}. Then Q_{i} ∈ Peel(D_{i}) if and only if Q_{i} ⊃ key(code(D_{i}^{w})) = K_{i}Y_{i} and [Q_{i} − (K_{i}Y_{i})] ∈ Peel(D_{i−1}). But Q_{i} has first block column X_{i}Y_{i} so by (8.11) it contains K_{i}Y_{i}. Also

[Q_{i} − (K_{i}Y_{i})] = [(X_{i} − K_{i})Y_{i}(W_{i−1}Y_{i−1}) ⋯ (W_{1}Y_{1})]
= [X_{i−1}Y_{i−1}(W_{i−2}Y_{i−2} ⋯ (W_{1}Y_{1})]
= Q_{i−1}

by (8.24) for i and i − 1, and (8.12) for i − 1. Since by induction Q_{i−1} ∈ Peel(D_{i−1}), we conclude that Q_{i} ∈ Peel(D_{i}). Taking i = n we have Q_{n} ∈ Peel(D_{n}).

The proof is complete because Q_{n} = Q and D_{n} = D. □

For the record, the next result is “rank stability for peelable tableaux”, which is analogous to the rank stability statements we proved for components of quiver degenerations (Proposition 4.13) and Zelevinsky pipe dreams (Proposition 5.15).

Corollary 8.21. There is a bijection Peel(D_{r}) → Peel(D_{m+r}) inducing a shape-preserving bijection Im(Ψ_{r}) → Im(Ψ_{m+r}).

Proof. With notation as in Definition 1.10, the diagrams D_{r}^{□} and D_{m+r}^{□} are the same up to removing some empty columns and rows. By Proposition 8.17 the images of Ψ_{r} and Ψ_{m+r} differ by a trivial relabeling of their entries. □

Corollary 8.21 could be used instead of Proposition 7.13 as the conduit through which Corollary 6.13 enters into the proof of Theorem 7.21.

8.4. The Buch–Fulton conjecture

Theorem 8.22. The map Ψ_{r} taking each D_{r}-peelable tableau Q ∈ Peel(D_{r}) to the list Ψ_{r}(Q) of tableaux obtained by deleting D_{Hom} from Q is a bijection from the set of D_{r}- peelable tableaux to r-factor sequences.

Proof. It suffices to show that the conditions on a tableau list (W_{1}, …, W_{n}) in Proposition 8.17 for membership in the image of Ψ_{r} are equivalent to the conditions defining an r-factor sequence. The case n = 1 is trivial, so suppose n ≥ 2. Write K̂_{i} to denote the analogue of the tableau K_{i} for r̂, for 1 ≤ i ≤ n − 1. Identifying the tableau array of r̂ with that of r but with T_{i} = T_{i,i−1} removed for all i, we have

(8.27)  K_{i} = [K̂_{i}T_{i}] for 1 ≤ i ≤ n,

where by convention

(8.28)  K̂_{n} = ∅.

Suppose first that (W_{1}, …, W_{n}) is an r-factor sequence. Let (U_{1}, …, U_{n−1}) be an r̂- factor sequence, with factorizations given by (8.3) and such that (8.4) and (8.5) hold. By induction the r̂-factor sequence (U_{1}, …, U_{n−1}) satisfies the conditions of Proposition 8.17 with associated tableaux X̂_{i} for 1 ≤ i ≤ n − 1. For convenience define

(8.29)  X̂_{n} = ∅

The X̂_{i} satisfy the following analogues of (8.12) through (8.21):

(8.30)  X̂_{i} = [(X̂_{i+1} − K̂_{i+1})U_{i}]
(8.31)  X̂_{i} ⊃ K̂_{i}  for 1 ≤ i ≤ n − 1
(8.32)  X̂_{1} = K̂_{1}.

Defining the tableau Ẑ_{i} by

(8.33)  Ẑ_{i} = [X̂_{i}U_{i−1} ⋯ U_{1}]_{B_{i}}

we have

(8.34)  height(Ẑ_{i}) ≤ r_{i−1,i}
(8.35)  width(Ẑ_{i}) ≤ r_{i,i+1}
(8.36)  wt(Ẑ_{i}) = wt(T_{≤i, ≥i}).

Note that the analogue X̂_{n−1} = U_{n−1} of (8.10) follows from (8.28), (8.29) and (8.30). Define X_{i} by

(8.37)  X_{i} = [X̂_{i} T_{i} P_{i−1}].

By (8.31) and (8.27) we have (8.11) and

(8.38)  [X_{i} − K_{i}] = [(X̂_{i} − K̂_{i})P_{i−1}].

We shall show by descending induction on i that X_{i} and Z_{i} (the latter being defined by (8.14)) have the properties specified by Proposition 8.17.

We have X_{n} = [∅T_{n}P_{n−1}] = [Q_{n}T_{n}P_{n−1}] = W_{n} by (8.29), (8.5), and (8.4), proving (8.10). Let n > i ≥ 1. We have

X_{i} = [X̂_{i}T_{i}P_{i−1}]
= [(X̂_{i+1} − K̂_{i+1})U_{i}T_{i}P_{i−1}]
= [(X̂_{i+1} − K̂_{i+1})P_{i}Q_{i}T_{i}P_{i−1}]
= [(X̂_{i+1} − K̂_{i+1})P_{i}W_{i}]
= [(X_{i+1} − K_{i+1})W_{i}]

by (8.37), (8.30), (8.3), (8.4), and (8.38), proving (8.12). We have X_{1} = [X̂_{1}T_{1}P_{0}] = [K̂_{1}T_{1}] = K_{1} by (8.37), (8.32), (8.5), and (8.27), proving (8.13).

By construction Z_{i} satisfies (8.17). To see this, observe first that ∑_{i=1}^{n} wt(W_{i}) is the weight of the entire tableau array. Then by (8.10) and (8.12), wt(X_{i}W_{i−1} ⋯ W_{1}) is the weight of the entire tableau array minus ∑_{j=i+1}^{m} wt(K_{j}), which is the part of the tableau array strictly west of the i^{th} block column. Finally, the restriction of X_{i}W_{i−1} ⋯ W_{1} to B_{i−1} restricts to the part of the tableau array weakly below the (i − 1)^{st} block row.

In particular, Z_{i} has at most r_{i} distinct letters appearing in it since the corresponding part of the tableau array has that property (see the proof of Proposition 8.17). Since Z_{i} is a tableau, (8.15) follows.

For (8.16), we have

(8.39)  \begin{aligned}Z_{i} & = & [(X̂_{i}T_{i}P_{i−1})(Q_{i−1}T_{i−1}P_{i−2}) ⋯ (Q_{1}T_{1}P_{0})]_{B_{i−1}} \\ & = & [X̂_{i}T_{i}U_{i−1}U_{i−2} ⋯ U_{1}]_{B_{i−1}} \end{aligned}

by (8.14), (8.37), (8.4), (8.3), (8.5), and the fact that T_{j} ∈ A._{j−1}. We get

(8.40)  \begin{aligned}Ẑ_{i−1} & = & [X̂_{i−1}U_{i−2} ⋯ U_{1}]_{B_{i−1}} \\ & = & [(X̂_{i} − K̂_{i})U_{i−1} ⋯ U_{1}]_{B_{i−1}} \\ & = & [(X̂_{i}T_{i} − K̂_{i}T_{i})U_{i−1} ⋯ U_{1}]_{B_{i−1}} \\ & = & [(X̂_{i}T_{i} − K_{i})U_{i−1} ⋯ U_{1}]_{B_{i−1}} \\ & = & [Z_{i} − K_{i}] \end{aligned}

by (8.33), (8.30), (8.27), and (8.39). Here we have used X̂_{i} ⊃ K̂_{i} so that [X̂_{i}T_{i}] ⊃ [K̂_{i}T_{i}] = K_{i}. By (8.40) we have

width(Ẑ_{i−1}) ≥ width(Z_{i}) − width(K_{i}).

But width(K_{i}) = width(T_{i}) = width(R_{i−1,i}) = r_{i} − r_{i−1,i}; together with (8.35) (applied to i − 1 instead of i), this implies (8.16).

Conversely, suppose (W_{1}, …, W_{n}) satisfies Proposition 8.17 with tableaux X_{i}. We must show that it is an r-factor sequence. By Lemma 8.18, conditions (8.18)–(8.21) hold. We claim that

(8.41)  W_{i} ⊃ T_{i}  for all 1 ≤ i ≤ n.

We have

(8.42)  X_{i} ⊃ T_{i}  for all 1 ≤ i ≤ n

by (8.11) and (8.1). For i = n, (8.10) and (8.42) gives (8.41). For n > i ≥ 1, (8.18) for i + 1, (8.12), and (8.42) imply (8.41).

Consider the factorization of W_{i} pictured below.

W_{i} =  T_{i} Q_{i} P_{i−1}

By construction (8.4) holds.

Since T_{n} has the largest letters, Q_{n} = ∅. To obtain P_{0} = ∅ (and finish up (8.5)), note that [K̂_{1}T_{1}] = K_{1} = X_{1} = [(X_{2} − K_{2})W_{1}] by (8.27), (8.13), and (8.12). But K_{1} has the same width as T_{1}. It follows that W_{1} does also, and hence that P_{0} = ∅.

Define U_{i} by (8.3). By induction it suffices to prove that there exist tableaux X̂_{i} such that (U_{1}, …, U_{n−1}) satisfies the conditions of Proposition 8.17. Define X̂_{n} by (8.29) and

(8.43)  X̂_{i} = [(X_{i+1} − K_{i+1})Q_{i}]  for 1 ≤ i ≤ n − 1.

We have

[X̂_{i}T_{i}P_{i−1}] = [(X_{i+1} − K_{i+1})Q_{i}T_{i}P_{i−1}] = [(X_{i+1} − K_{i+1})W_{i}] = X_{i}

by (8.43), (8.4), and (8.12), proving (8.37). This implies (8.31). We get

X̂_{i} = [(X_{i+1} − K_{i+1})Q_{i}]
= [(X̂_{i+1}T_{i+1}P_{i} − K̂_{i+1}T_{i+1})Q_{i}]
= [(X̂_{i+1} − K̂_{i+1})P_{i}Q_{i}]
= [(X̂_{i+1} − K̂_{i+1})U_{i}]

by (8.43), by (8.37), (8.27), and (8.31) for i+1 (which hold by induction), and (8.3), proving (8.30). We have [K̂_{1}T_{1}] = K_{1} = X_{1} = [X̂_{1}T_{1}P_{0}] = [X̂_{1}T_{1}] by (8.27), (8.13), (8.37), and (8.5), from which (8.32) follows.

Define Z_{i} by (8.14) and Ẑ_{i} by (8.33). Note that (8.36) holds by construction. This implies (8.34) because Ẑ_{i} is a tableau in an alphabet with at most r_{i,i+1} distinct letters. To show that (8.35) holds, note that (8.40) still holds. Let D_{i,j} be as in the proof of Proposition 8.17. Recall that also that Z_{i} ∈ Peel(D_{i,i−1}) by (8.26). Apply Lemma 8.15 with D_{i,i−1}^{w} given by the west block column of D_{i,i−1}. Note that K_{i} = key(code(D_{i,i−1}^{w})). It follows that the r_{i}-fold peeling of Z_{i} (which is none other than Ẑ_{i} by (8.40)) is D_{i,i}-peelable. The inequality (8.35) follows. Since the conditions of Proposition 8.17 have been satisfied, (U_{1}, …, U_{n−1}) is in the image of Ψr̂. By induction (U_{1}, …, U_{n−1}) is an r̂-factor sequence. Thus (W_{1}, …, W_{n}) is an r-factor sequence. □

Our final result is an alternate combinatorial version of the formula in Theorem 7.21. The list of shapes in a tableau list is called the shape of the tableau list.

Corollary 8.23 (Buch–Fulton conjecture [BF99]). The quiver constant c_{\underline{λ}}(r) in the expression Q_{r}(x − ^{∘}x) = ∑_{\underline{λ}} c_{\underline{λ}}(r)s_{\underline{λ}}(x_{r} − ^{∘}x_{r}) of the quiver polynomial in terms of Schur functions (or by Theorem 7.10, in the expression of double quiver functions in terms of double Schur functions) equals the number of r-factor sequences of shape \underline{λ}.

Proof. Theorem 8.22 and Theorem 7.21. □

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Mathematics Department, UC Berkeley, Berkeley, California

   E-mail address: allenk@math.berkeley.edu

Mathematical Sciences Research Institute, Berkeley, California

   E-mail address: ezra@math.umn.edu

Mathematics Department, Virginia Tech, Blacksburg, Virginia

   E-mail address: mshimo@math.vt.edu