On Horn's Problem and its Volume Function

Robert Coquereaux (1, 2) , Colin Mcswiggen (3, 4) , Jean-Bernard Zuber (4)

1 CPT - Centre de Physique Théorique - UMR 7332
2 CPT - E2 Géométrie, Physique et Symétries
3 Brown University
4 LPTHE - Laboratoire de Physique Théorique et Hautes Energies

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).

Domaines: Physique mathématique [math-ph]; Probabilités [math.PR]; Théorie des représentations [math.RT]

Robert Coquereaux, Colin Mcswiggen, Jean-Bernard Zuber. On Horn's Problem and its Volume Function. Communications in Mathematical Physics, 2020, 376 (3), pp.2409-2439. ⟨10.1007/s00220-019-03646-7⟩. ⟨hal-02089584⟩