Projections of orbital measures, Gelfand-Tsetlin polytopes, and splines

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The unitary group U(N) acts by conjugations on the space H (N) of N × 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 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 × K, where K = 2; : : : ;N - 1, then the pushforward of a generic orbital measure is still computable: its density is given by a K × K determinant composed from one-dimensional splines. The result can also be reformulated in terms of projections of the Gelfand-Tsetlin polytopes.

Original languageEnglish
Pages (from-to)1011-1022
Number of pages12
JournalJournal of Lie Theory
Issue number4
Publication statusPublished - 2013
Externally publishedYes


  • B-spline
  • Gelfand-Tsetlin polytope
  • Harish-Chandra-Itzykson- Zuber integral
  • Orbital measure


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