Generalization of the Macdonald formula for Hall-Littlewood polynomials

TL;DR

This paper develops combinatorial tools to describe the Gaussent-Littelmann formula for Hall-Littlewood polynomials across types A_n, B_n, and C_n, and demonstrates its equivalence to the Macdonald formula for type A_n.

Contribution

It introduces a combinatorial framework linking Young tableaux with the Gaussent-Littelmann formula and proves their equivalence to the Macdonald formula for type A_n.

Findings

Unified combinatorial description for types A_n, B_n, C_n

Equivalence of Gaussent-Littelmann and Macdonald formulas for type A_n

Young tableaux as a key tool in the description

Abstract

We study the Gaussent-Littelmann formula for Hall-Littlewood polynomials and we develop combinatorial tools to describe the formula in a purely combinatorial way for type A_n, B_n and C_n. This description is in terms of Young tableaux and arises from identifying one-skeleton galleries that appear in the Gaussent-Littelmann formula with Young tableaux. Furthermore, we show by using these tools that the Gaussent-Littelmann formula and the well-known Macdonald formula for Hall-Littlewood polynomials for type A_n are the same.