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

Abstract

Horn's problem, i.e., the study of the eigenvalues of the sum C = A + B of two matrices, given the spectrum of A and of B, is re-examined, comparing the case of real symmetric, complex Hermitian and self-dual quaternionic 3×3 matrices. In particular, what can be said on the probability distribution function (PDF) of the eigenvalues of C if A and B are independently and uniformly distributed on their orbit under the action of, respectively, the orthogonal, unitary and symplectic group? While the two latter cases (Hermitian and quaternionic) may be studied by use of explicit formulae for the relevant orbital integrals, the case of real symmetric matrices is much harder. It is also quite intriguing, since numerical experiments reveal the occurrence of singularities where the PDF of the eigenvalues diverges. Here we show that the computation of the PDF of the symmetric functions of the eigenvalues for traceless 3×3 matrices may be carried out in terms of algebraic functions -roots of quartic polynomials -and their integrals. The computation is carried out in detail in a particular case, and reproduces the expected singular patterns. The divergences are of logarithmic or inverse power type. We also relate this PDF to the (rescaled) structure constants of zonal polynomials and introduce a zonal analogue of the Weyl SU(n) characters.

On Horn’s Problem and Its Volume Function

“…This paper discusses several constructions related to the extended Horn's problem and to tensor product multiplicities, including a number of previously known results that we recall for the sake of completeness. However, to the authors' knowledge, Propositions 1 through 4, equation (19), and the conjectured expression (59) either are new or extend in various ways several results previously obtained by two of us in [42,7,8]. Proposition 3 is a particular instance of a more general phenomenon whereby the volumes of certain symplectic manifolds equal the volumes of polytopes whose integer points count representation multiplicities [17,16].…”

“…Next, in sect. 2.2 we relate J to the Riemannian geometry of the orbits, considered as submanifolds of V , and we explain how this interpretation helps to understand the nature of the divergences that arise in J in the case of real symmetric matrices [8].…”

Revisiting Horn’s problem

“…In the orthogonal (θ " 1 2 ) case, most unfortunately, there is no similar compact expression. The best that may be achieved is a series expansion in terms of zonal polynomials (see [5] and references therein), which is not very handy for detailed calculations.…”

“…7, and inverse-square-root divergences at the two special points pγ 1 , γ 2 q " p1, 0q and p2, 0q. Note that because of the vanishing of the Vandermonde determinant, ppγ 1 , γ 2 q may have a smaller locus of singularity than ρ, see [5].…”

“…where the sum on the right runs over a finite, pλ, µ, νq-independent set of weights K, with nonnegative coefficients r κ satisfying [5] ÿ κ r κ dim V κ " 1 .…”