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

2010/04/01 by Stéphane Gaussent, Gaussent, Stéphane, Peter Littelmann +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1004.0066

openalex publication_date 2010/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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