Revisiting Horn's problem

Robert Coquereaux (1, 2) , Colin Mcswiggen (3) , Jean-Bernard Zuber (4)

1 CPT - E2 Géométrie, Physique et Symétries
2 CPT - Centre de Physique Théorique - UMR 7332
3 Brown University
4 LPTHE - Laboratoire de Physique Théorique et Hautes Energies

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.

Domaines: Physique mathématique [math-ph]

Robert Coquereaux, Colin Mcswiggen, Jean-Bernard Zuber. Revisiting Horn's problem. Journal of Statistical Mechanics: Theory and Experiment, 2019, Special Issue in Memory of Vladimir Rittenberg, 2019 (9), pp.094018. ⟨10.1088/1742-5468/ab3bc2⟩. ⟨hal-02418056⟩