One-skeleton galleries, the path model, and a generalization of Macdonald’s formula for Hall-Littlewood polynomials. (English) Zbl 1244.05226
Summary: We give a direct geometric interpretation of the path model using galleries in the 1-skeleton of the Bruhat-Tits building associated with 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 \(A_{n}\) turns out to be equivalent to Macdonald’s formula.
MSC:
05E05 | Symmetric functions and generalizations |
05E10 | Combinatorial aspects of representation theory |
22E46 | Semisimple Lie groups and their representations |