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

TL;DR

This paper investigates the eigenvalue distribution of the sum of two matrices in different symmetry classes, revealing singularities and providing explicit algebraic formulas, especially for real symmetric matrices, and relating these to zonal polynomials.

Contribution

It develops explicit algebraic methods to compute the eigenvalue PDFs for traceless 3x3 real symmetric matrices, including singularity analysis, and introduces zonal analogues of Weyl characters.

Findings

Eigenvalue PDFs exhibit logarithmic and inverse power divergences.

Explicit algebraic formulas involve roots of quartic polynomials.

The approach links eigenvalue distributions to zonal polynomials and their structure constants.

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 $3\times 3$ 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 $3\times 3$ 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 ${\rm SU}(n)$ characters.