One-skeleton galleries, the path model and a generalization of Macdonald's formula for Hall-Littlewood polynomials

TL;DR

This paper provides a geometric interpretation of the path model using galleries in the Bruhat-Tits building, enabling computation of Hall-Littlewood polynomial coefficients and generalizing Macdonald's formula.

Contribution

It introduces a direct geometric approach to the path model, leading to a new formula for Hall-Littlewood polynomials that generalizes Macdonald's classical result.

Findings

Derived a geometric formula for Hall-Littlewood coefficients

Connected the path model to Macdonald's formula in type A

Provided a 'geometric compression' of Schwer's formula

Abstract

We give a direct geometric interpretation of the path model using galleries in the $1-$skeleton of the Bruhat-Tits building associated to a semi-simple algebraic group. This interpretation allows us to compute the coefficients of the expansion of the Hall-Littlewood polynomials in the monomial basis. The formula we obtain is a "geometric compression" of the one proved by Schwer, its specialization to the case ${\tt A}_n$ turns out to be equivalent to Macdonald's formula.