Brion’s theorem for Gelfand–Tsetlin polytopes

Abstract

This work is motivated by the observation that the character of an irreducible gl _n -module (a Schur polynomial), being the sum of exponentials of integer points in a Gelfand–Tsetlin polytope, can be expressed by using Brion’s theorem. The main result is that, in the case of a regular highest weight, the contributions of all nonsimplicial vertices vanish, while the number of simplicial vertices is n! and the contributions of these vertices are precisely the summands in Weyl’s character formula.