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