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

Stéphane Gaussent (1) , Peter Littelmann (2)

1 IECN - Institut Élie Cartan de Nancy
2 Mathematisches Institut, Universität zu Köln

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.

Domaines: Théorie des représentations [math.RT]; Combinatoire [math.CO]

Stéphane Gaussent, Peter Littelmann. One-skeleton galleries, the path model and a generalization of Macdonald's formula for Hall-Littlewood polynomials. International Mathematics Research Notices, 2012, 12, pp.2649-2707. ⟨hal-00468851v2⟩