On Horn's Problem and its Volume Function

Abstract : We consider an extended version of Horn's problem: given two orbits Oα and Oβ of a linear representation of a compact Lie group, let A in Oα , B in Oβ be independent and invariantly distributed random elements of the two orbits. The problem is to describe the probability distribution of the orbit of the sum A+B. We study in particular the familiar case of coadjoint orbits, and also the orbits of self-adjoint real, complex and quaternionic matrices under the conjugation actions of SO(n), SU(n) and USp(n) respectively. The probability density can be expressed in terms of a function that we call the volume function. In this paper, (i) we relate this function to the symplectic or Riemannian geometry of the orbits, depending on the case; (ii) we discuss its non-analyticities and possible vanishing; (iii) in the coadjoint case, we study its relation to tensor product multiplicities (generalized Littlewood-Richardson coefficients) and show that it computes the volume of a family of convex polytopes introduced by Berenstein and Zelevinsky. These considerations are illustrated by a detailed study of the volume function for the coadjoint orbits of B2=so(5).
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Contributor : Robert Coquereaux <>
Submitted on : Wednesday, April 3, 2019 - 10:04:22 PM
Last modification on : Wednesday, April 17, 2019 - 1:44:39 AM


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  • HAL Id : hal-02089584, version 1
  • ARXIV : 1904.00752


Robert Coquereaux, Colin Mcswiggen, Jean-Bernard Zuber. On Horn's Problem and its Volume Function. 2019. ⟨hal-02089584⟩



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