Hall-littlewood Vertex Operators and Generalized Kostka Polynomials Mark Shimozono and Mike Zabrocki

Kostka-Folkes polynomials may be considered as coefficients of the formal power series representing the character of certain graded GL(n)-modules. These GL(n)-modules are defined by twisting the coordinate ring of the nullcone by a suitable line bundle [1] and the definition may be generalized by twisting the coordinate ring of any nilpotent conjugacy closure in gl(n) by a suitable vector bundle [13]. The resulting polynomials have been called generalized Kostka polynomials. Jing defined a vertex operator that generates the Hall-Littlewood symmetric function Q[X; q] [6], thereby giving an elegant symmetric function recursion for the Kostka-Foulkes polynomials. Garsia used a variant of Jing’s vertex operator to derive various new formulas for the Kostka-Foulkes polynomials. Our point of departure was the observation that the Hall-Littlewood vertex operators can be used to obtain formulas for generalized Kostka polynomials. Our treatment uses Garsia’s plethystic type formulas. One striking fact is that the Z[q, q−1]-linear span of n-fold compositions of components of the Hall-Littlewood vertex operators, is isomorphic to KG×C∗(N ), the GL(n) × C ∗-equivariant K-theory of the nullcone. Under this isomorphism, an n-fold composite operator is sent to the class of the Euler characteristic of a twisted module. This fact has a generalization for all the nilpotent conjugacy class closures in gl(n). These Grothendieck groups were studied in [7]. We derive many explicit relations among the vertex operators, most of which can be in interpreted as relations in the Grothendieck groups which arise from certain Koszul complexes. This allows for more explicit proofs of some basis theorems for these Grothendieck groups that were proved in [7] using geometric arguments. There is a particularly well-behaved subfamily of the generalized Kostka polynomials, namely, those that are indexed by a sequence of rectangular partitions. For this subfamily almost all of the formulas for Kostka-Foulkes polynomials have generalizations. There are combinatorial formulas involving Littlewood-Richardson tableaux [11] [10], rigged configurations [8], and inhomogeneous paths with energy function [12] [10].