On Horn’s Problem and Its Volume Function

Abstract

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 P O α , B P 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 SOpnq, SUpnq and USppnq 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 " sop5q.

Revisiting Horn’s problem

“…For each such term, the integral over T thus yields the multiplicity λ b µ b κ Ñ ν. There is a similar identity involving J at unshifted weights pλ, µ, νq [7]. We conclude that for a triple pλ, µ, νq such that λ`µ´ν P Q, we have two identities (distinct if ρ R Q):…”

“…Complex Hermitian SUpnq 2 Quaternionic Self-Dual USppnq For given n and α, β, not only the support of the distribution of γ is the same [12], but also the locus of singularity of the PDF (although the singularities themselves are of quite a different nature). We may state the following proposition [7]:…”

“…Hint at proof: Consider the map Φ : GˆG Ñ M θ n , pg 1 , g 2 q Þ Ñ C " A`B " g 1 αg´1 1`g 2 βg´1 2 . If C is a regular value of Φ in the sense that the differential dΦ is surjective at all points of the preimage Φ´1pCq, then the PDF is real analytic at γ. Non-analyticities can therefore only occur at γ that are spectra of non-regular values of Φ, and it is easy to see that these lie on hyperplanes of the form (2), see details in [7].…”

“…Using standard results in combinatorics, it can be shown that the leading coefficient a d of P ν λµ is a constant equal to the volume of H ν λµ . (In fact it is the relative volume, given by the Euclidean d-volume times a scalar factor that can be computed, see [7] for details.) We now want to show that this volume is given by the function J .…”

The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices

“…That non C ∞ differentiability occurs only on these lines has now been established in full generality, following an argument of M. Vergne, see[7].…”