Revisiting Horn’s problem

Abstract

We review recent progress on Horn's problem, which asks for a description of the possible eigenspectra of the sum of two matrices with known eigenvalues.After revisiting the classical case, we consider several generalizations in which the space of matrices under study carries an action of a compact Lie group, and the goal is to describe an associated probability measure on the space of orbits. We review some recent results about the problem of computing the probability density via orbital integrals and about the locus of singularities of the density. We discuss some relations with representation theory, combinatorics, pictographs and symmetric polynomials, and we also include some novel remarks in connection with Schur's problem.

Box splines, tensor product multiplicities and the volume function

“…The volume function is of independent significance in symplectic geometry, as well as in random matrix theory, where it is closely related to the joint spectral density for the randomized Horn's problem; see (6) below. All of the main results involving J in this paper can be reformulated in terms of this probability density, so that in a loose sense this paper can be interpreted as establishing an equivalence between two problems: the problem of computing tensor product multiplicities for a compact semisimple Lie algebra g, and the randomized Horn's problem for coadjoint orbits in g * .…”

“…In addition to this probabilistic interpretation, J (α, β; γ) can also be interpreted geometrically both as the volume of a symplectic reduction of the product of coadjoint orbits O α × O β × O −γ and as the volume of a convex polytope constructed by Berenstein and Zelevinsky [1], the integer points of which count tensor product multiplicities. We refer the reader to [6,7,9] for further background on the volume function and for details of its probabilistic and geometric interpretations.…”

Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform

“…In particular for β = 2, the Heckman-Opdam hypergeometric function admits a multiplicative counterpart of the Itzykson-Zuber determinantal formula (8), known as the Gelfand-Naimark [23] formula:…”

Rank one HCIZ at high temperature: interpolating between classical and free convolutions

“…where ∆(a) := i< j (a i − a j ), is the Vandermonde determinant. The HCIZ integral has applications in problems directly linked to random matrix theory (RMT) such as the study of the sum of invariant ensembles [22][23][24], the development of large deviation principles [25], the study of the so-called orbital beta processes [26]. It is also linked to the enumeration of Hurwitz numbers in algebraic geometry [27,28] and to quantum ergodic transport ( [29]), to cite a few recent results.…”