Grothendieck classes of quiver varieties

Abstract

Abstract. We prove a formula for the structure sheaf of a quiver variety in the Grothendieck ring of its embedding variety. This formula generalizes and gives new expressions for Grothendieck polynomials. We furthermore conjecture that the coefficients in our formula have signs which alternate with degree. The proof of our formula involves K-theoretic generalizations of several useful cohomological tools, including the Thom-Porteous formula, the Jacobi-Trudi formula, and a Gysin formula of Pragacz.

Alternating formulas for K -theoretic quiver polynomials

“…A crucial property of arbitrary stable double Grothendieck polynomials, proved in [Buc02b, Theorem 6.13], is that every such polynomialĜ w (z/ż) has a unique expression is a sum of products of stable double Grothendieck polynomialsĜ µ j (x j−1 /x j ) for grassmannian permutations µ j , with uniquely determined integer coefficients c µ (r). That these coefficients are the same as in Corollary 16 follows from the fact that the right side above determines the same element in the n th tensor power of Buch's bialgebra Γ from [Buc02b,Buc02a] as does the right side of the top formula in Corollary 16.…”

“…This K-theoretic version is still in terms of lacing diagrams, but nonminimal diagrams contribute terms of higher degree. The motivating consequence is a conjecture of Buch on the sign-alternation of the coefficients appearing in his expansion of quiver K-polynomials in terms of stable Grothendieck polynomials for partitions [Buc02a]. …”

A Littlewood-Richardson rule for the K-theory of Grassmannians

“…We hope that the statements in the last two sections will be comprehensible after reading w and the first seven lines of w This paper came out of a project aimed at finding a formula for the structure sheaf of a quiver variety. We will present such a formula in [2], thus generalizing our earlier results with W. Fulton regarding the cohomology class of a quiver variety [3]. The proof A.S. BUCH of the cohomology formula was relatively simple, because some powerful cohomological tools related to the ring of symmetric functions were already available.…”

A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus

“…We believe that further developing this connection may allow one to, for example, prove a K-theory analogue of the "factor sequence formula" conjectured in [Buch and Fulton 1999] and proved in [Knutson et al 2006], which is a problem that has remained open in this topic; see [Buch 2002a;2005a]. (In [Buch et al 2008] a different factor sequence formula, generalizing the one given in [Buch 2005a], was given.…”

“…Indeed, there has been significant interest in the Grothendieck ring of X and of related varieties; see work on, for example, quiver loci [Buch 2002a;2005a;Buch et al 2008], Hilbert series of determinantal ideals [Knutson and Miller 2005;Knutson et al 2008;2009], applications to invariants of matroids [Speyer 2006], and in relation to representation theory [Griffeth and Ram 2004;Lenart and Postnikov 2007;Willems 2006]. See also work of concerning combinatorial Hopf algebras.…”