Four positive formulae for type A quiver polynomials

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 was conjectured for by 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 versions, 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 [Jos84, 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 schemetheoretic isomorphisms (originally proved in [LM98] using standard monomial theory). 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 double quiver functions 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 a 'double stable' generalization of the Buch-Fulton conjecture.

Alternating signs of quiver coefficients

“…The component formulas are proved using a combination of the Gröbner degeneration and the ratio formula, as well as a combinatorial study of a double version of the ratio formula. In particular, it is proved that a limit of the double ratio formula agrees with the double quiver functions introduced in [10,7] and named in [24]. The authors of [24] have informed us that they can generalize their methods to work in K-theory, although, according to their own description, this approach is rather complicated.…”

“…In their recent paper [24], Knutson, Miller, and Shimozono deliver a breakthrough within the theory of quiver formulas and prove at least two explicit combinatorial formulas for the cohomological quiver coefficients, which show that these coefficients are non-negative. One of the important ideas in their work is to reinterpret the lace diagrams of Abeasis and Del Fra [1] as sequences of partial permutations.…”

“…This interpretation is explained by a Gröbner degeneration of a quiver variety in a matrix space into a union of products of matrix Schubert varieties. The component formula of [24] writes the cohomology class of a quiver variety as a sum, over all 'minimal ' lace diagrams, of the products of the Schubert polynomials for the corresponding partial permutations. The proof that quiver coefficients are nonnegative is obtained by proving a stable version of this component formula, where the Schubert polynomials are replaced with Stanley symmetric functions.…”

“…A special case of this conjecture, corresponding to a particular choice of tableaux diagram, was proved in [24]. However, so far there has been no progress in generalizing this conjecture to K-theory.…”

“…Starting from the ratio formula, we give combinatorial proofs of K-theoretic generalizations of the component formulas, where the Schubert polynomials and Stanley symmetric functions are replaced with ordinary and stable Grothendieck polynomials. These formulas are given in terms of sequences of partial permutations, which we call KMS-factorizations of the Zelevinsky permutation [34,24]. To conclude that K-theoretic quiver coefficients have alternating signs and to derive the combinatorial formula for these coefficients, we use Lascoux's result that stable Grothendieck polynomials are linear combinations with alternating signs of stable Grothendieck polynomials for partitions [27].…”