Graded characters of modules supported in the closure of a nilpotent conjugacy class

TL;DR

This paper investigates graded GL(n)-modules supported in nilpotent conjugacy class closures, deriving properties and formulas for their Poincare polynomials, which generalize Kostka-Foulkes and relate to Littlewood-Richardson coefficients.

Contribution

It introduces a new family of polynomials generalizing Kostka-Foulkes, with formulas like Morris recurrence and q-Kostant, and conjectures involving tableaux and charge.

Findings

Derived properties and formulas for the polynomials

Established a generalized Morris recurrence

Proposed a conjectural formula involving tableaux

Abstract

We study the Poincare polynomials of isotypic components of a natural family of graded GL(n)-modules supported in the closure of a nilpotent conjugacy class. These polynomials generalize the Kostka-Foulkes and are q-analogues of Littlewood-Richardson coefficients corresponding to arbitrary tensor products of irreducibles. Many properties and formulas for these polynomials are derived, such as a generalized Morris recurrence, q-Kostant formula, and a conjectural formula in terms of catabolizable tableaux and charge.