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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 aGelfand-Tsetlin polytope and can thus be calculated via Brion's theorem. Inorder to show how the result of such a calculation matches Weyl's characterformula we prove some interesting combinatorial traits of Gelfand-Tsetlinpolytopes. Namely, we show that under the relevant substitution the integerpoint transforms of all but $n!$ vertices vanish, the remaining ones being thesummands in Weyl's formula.

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