On Horn’s Problem and Its Volume Function

We consider an extended version of Horn’s problem: given two orbits O α and O β of a linear representation of a compact Lie group, let A ∈ O α , B ∈ O β 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 B 2 = so ( 5 ) .