A generalization of the Kostka-Foulkes polynomials

TL;DR

This paper explores a broad generalization of Kostka-Foulkes polynomials using rigged configurations, connecting combinatorial objects to representation theory and conjecturing their equivalence to Poincare polynomials of specific graded modules.

Contribution

It introduces a new generalization of Kostka-Foulkes polynomials through rigged configurations and links them to geometric and representation-theoretic structures.

Findings

Kostka-Foulkes and Macdonald-Kostka polynomials are special cases.

Conjecture that these polynomials match Poincare polynomials of certain graded modules.

Provides a combinatorial framework for understanding these polynomials.

Abstract

Combinatorial objects called rigged configurations give rise to q-analogues of certain Littlewood-Richardson coefficients. The Kostka-Foulkes polynomials and two-column Macdonald-Kostka polynomials occur as special cases. Conjecturally these polynomials coincide with the Poincare polynomials of isotypic components of certain graded GL(n)-modules supported in a nilpotent conjugacy class closure in gl(n).