Projections of Orbital Measures, Gelfand-Tsetlin Polytopes, and Splines

G. Olshanski

Journal of Lie theory, Tome 23 (2013) no. 4, pp. 1011-1022

Résumé

The unitary group $U(N)$ acts by conjugations on the space ${\cal H}(N)$ of $N\times N$ Hermitian matrices, and every orbit of this action carries a unique invariant probability measure called an orbital measure. Consider the projection of the space ${\cal H}(N)$ 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 $K\times K$, where $K=2,\dots,N-1$, then the pushforward of a generic orbital measure is still computable: its density is given by a $K\times K$ determinant composed from one-dimensional splines. The result can also be reformulated in terms of projections of the Gelfand-Tsetlin polytopes.

Détail

Classification : 22E30 41A15
Mots-clés : Orbital measure, Gelfand-Tsetlin polytope, B-spline, Harish-Chandra-Itzykson-Zuber integral