Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 029, 34 pages      arXiv:1809.03394      https://doi.org/10.3842/SIGMA.2019.029

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

Robert Coquereaux a and Jean-Bernard Zuber b
a) Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France
b) Sorbonne Université, UMR 7589, LPTHE, F-75005, Paris, France CNRS, UMR 7589, LPTHE, F-75005, Paris, France

Received December 20, 2018, in final form April 06, 2019; Published online April 16, 2019

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.

Key words: Horn problem; honeycombs; polytopes; zonal polynomials; Littlewood-Richardson coefficients.

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