This article is cited in 4 papers

Brion's Theorem for Gelfand–Tsetlin Polytopes

I. Yu. Makhlin<sup>ab

a</sup> L.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences
^b International Laboratory of Representation Theory and Mathematical Physics, National Research University Higher School of Economics

Abstract: 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.

UDC: 512.815.1

Received: 15.10.2015

DOI: 10.4213/faa3232

 English version: Functional Analysis and Its Applications, 2016, 50:2, 98–106

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