Weyl's Formula as the Brion Theorem for Gelfand-Tsetlin Polytopes

Abstract

We exploit the idea that the character of an irreducible finite dimensional ${\mathfrak{gl}}_n$-module is the sum of certain exponents of integer points in a Gelfand-Tsetlin polytope and can thus be calculated via Brion's theorem. In order to show how the result of such a calculation matches Weyl's character formula we prove some interesting combinatorial traits of Gelfand-Tsetlin polytopes. Namely, we show that under the relevant substitution the integer point transforms of all but $n!$ vertices vanish, the remaining ones being the summands in Weyl's formula.

References in corpus (1)

• Characters of Feigin-Stoyanovsky subspaces and Brion's theorem

Cited by in corpus (2)

• Plane partitions with a "pit": generating functions and representation theory
• Vertices of FFLV polytopes