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

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.