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

References in corpus (5)

• On Horn's Problem and its Volume Function
• WKB-expansion of the HarishChandra-Itzykson-Zuber integral for arbitrary beta
• On some properties of SU(3) Fusion Coefficients
• On Products of Delta Distributions and Resultants
• Diagonals of real symmetric matrices of given spectra as a measure space

Cited by in corpus (8)

• On Horn's Problem and its Volume Function
• Large Deviation Principles via Spherical Integrals
• Rank one HCIZ at high temperature: interpolating between classical and free convolutions
• Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform
• Co-rank 1 projections and the randomised Horn problem
• Projections of Orbital Measures and Quantum Marginal Problems
• On Products of Delta Distributions and Resultants
• Multiplicities, pictographs, and volumes