Brion's Theorem for Gelfand–Tsetlin Polytopes

This work is motivated by the observation that the character of an irreducible $\mathfrak{gl}_n$-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.