Geodesic
Brion's Theorem for Gelfand--Tsetlin Polytopes
Funkcionalʹnyj analiz i ego priloženiâ, Tome 50 (2016) no. 2, pp. 20-30

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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.
Keywords: Gelfand–Tsetlin polytopes, Brion's theorem, Schur polynomials, general linear Lie algebra. 
@article{FAA_2016_50_2_a1,
     author = {I. Yu. Makhlin},
     title = {Brion's {Theorem} for {Gelfand--Tsetlin} {Polytopes}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {20--30},
     publisher = {mathdoc},
     volume = {50},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2016_50_2_a1/}
}
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I. Yu. Makhlin. Brion's Theorem for Gelfand--Tsetlin Polytopes. Funkcionalʹnyj analiz i ego priloženiâ, Tome 50 (2016) no. 2, pp. 20-30. http://geodesic.mathdoc.fr/item/FAA_2016_50_2_a1/