Some aspects of Horn's problem

by Prof. Jean-Bernard Zuber (Sorbonne Université, Laboratoire de Physique Théorique et Hautes Énergies)




Horn's problem deals with the following question: what can be said about the spectrum of eigenvalues of two Hermitian matrices of given spectrum ? The support of the spectrum  is now well understood, after a long series of works from Weyl (1912)  to Horn (1952) to Klyachko (1998) and Knutson and Tao (1999). The problem has also amazing connections with group  theory and the decomposition of tensor product of representations. Comparison with the same problem for real symmetric matrices and the action of the orthogonal group reveals similarities but also unexpected differences… In this talk, after a short introduction to the problem, I'll  sketch the computation of  the probability distribution function of the eigenvalues. I'll also review some  aspects of the connection with representation theory and combinatorics.