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

Authors
  • Makhlin, Igor
Type
Published Article
Publication Date
May 18, 2016
Submission Date
Sep 29, 2014
Identifiers
arXiv ID: 1409.7996
Source
arXiv
License
Yellow
External links

Abstract

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 a Gelfand-Tsetlin polytope and can thus be calculated via Brion's theorem. In order to show how the result of such a calculation matches Weyl's character formula we prove some interesting combinatorial traits of Gelfand-Tsetlin polytopes. Namely, we show that under the relevant substitution the integer point transforms of all but $n!$ vertices vanish, the remaining ones being the summands in Weyl's formula.

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