Double Kostka polynomials and Hall bimodule

TL;DR

This paper explores the combinatorial properties of double Kostka polynomials and Hall-Littlewood functions, establishing key formulas and an isomorphism with symmetric functions involving two variable types.

Contribution

It demonstrates a Lascoux-Schutzenberger type formula for double Kostka polynomials and proves the Hall bimodule's isomorphism to a ring of symmetric functions with two variable types.

Findings

Lascoux-Schutzenberger type formula holds in certain cases

Hall bimodule is isomorphic to a ring of symmetric functions with two variables

Provides an alternative approach to existing results

Abstract

Double Kostka polynomials are polynomials indexed by a pair of double partitions. As in the ordinary case, double Kostka polynomials are defined in terms of Schur functions and Hall-Littlewood functions associated to double partitions. In this paper, we study combinatorial properties of those double Kostka polynomials and Hall-Littlewood functions. In particular, we show that the Lascoux-Schutzenberger type formula holds for double Kostka polynomials in certain cases. Moreover, we show that the Hall bimodule introduced by Finkelberg-Ginzburg-Travkin is isomorphic to the ring of symmetric functions with two types of variables, which gives an alternate approach for their result.