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29 March 2024
 
  » arxiv » 1904.0752

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On Horn's Problem and its Volume Function
Robert Coquereaux ; Colin McSwiggen ; Jean-Bernard Zuber ;
Date 1 Apr 2019
AbstractWe consider an extended version of Horn’s problem: given two orbits $mathcal{O}_alpha$ and $mathcal{O}_eta$ of a linear representation of a compact Lie group, let $Ain mathcal{O}_alpha$, $Bin mathcal{O}_eta$ 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 $mathrm{SO}(n)$, $mathrm{SU}(n)$ and $mathrm{USp}(n)$ 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 $B_2=mathfrak{so}(5)$.
Source arXiv, 1904.0752
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