Projections of orbital measures, Gelfand-Tsetlin polytopes, and splines
Abstract
The unitary group U(N) acts by conjugations on the space H(N) of NxN Hermitian matrices, and every orbit of this action carries a unique invariant probability measure called an orbital measure. Consider the projection of the space H(N) onto the real line assigning to an Hermitian matrix its (1,1)-entry. Under this projection, the density of the pushforward of a generic orbital measure is a spline function with N knots. This fact was pointed out by Andrei Okounkov in 1996, and the goal of the paper is to propose a multidimensional generalization. Namely, it turns out that if instead of the (1,1)-entry we cut out the upper left matrix corner of arbitrary size KxK, where K=2,...,N-1, then the pushforward of a generic orbital measure is still computable: its density is given by a KxK determinant composed from one-dimensional splines. The result can also be reformulated in terms of projections of the Gelfand-Tsetlin polytopes.
Keywords
Cite
@article{arxiv.1302.7116,
title = {Projections of orbital measures, Gelfand-Tsetlin polytopes, and splines},
author = {Grigori Olshanski},
journal= {arXiv preprint arXiv:1302.7116},
year = {2013}
}
Comments
12 pages; to appear in Journal of Lie Theory