Weyl's Formula as the Brion Theorem for GelfandTsetlin Polytopes
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 GelfandTsetlin 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 GelfandTsetlin 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2014
 arXiv:
 arXiv:1409.7996
 Bibcode:
 2014arXiv1409.7996M
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Combinatorics
 EPrint:
 Brion's theorem for GelfandTsetlin polytopes, Functional Analysis and Its Applications, 50:2 (2016), pp 98106