2013/02/28 by Grigori Olshanski, Olshanski, Grigori · 2 citations
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Representation Theory (math.RT) #math.CA #math.RT
paper · pdf · doi:10.48550/arxiv.1302.7116
12 pages; to appear in Journal of Lie Theory
arxiv created 2013/02/28 · arxiv updated 2013/03/04
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.