Algebraic structures associated to group actions and sums of Hermitian matrices (Q2740886)

The aim of these lecture notes is to show how basic ideas about hypergroups may help to understand the distribution of the eigenvalues of sums of two independent randomly chosen Hermitian matrices with given eigenvalues. For this the author first gives a low level introduction to commutative hypergroups and, in particular, the example appearing as orbit hypergroups. The author then focuses on the orbit hypergroup \(K\) which appears when \(SU(n)\) acts on the additive group \(H(n)\) of all Hermitian matrices by conjugation. Here, \(K\) may be regarded as the closure \(\overline C\) of some Weyl chamber \(C\subset\mathbb{R}^n\) associated with the Weyl group \(S_n\), and the distribution of the sum of two random Hermitian matrices with given eigenvalues is just the convolution product of the associated point measures. These distributions are now described more precisely by using the method of sums of adjoint orbits developed by A. H. Dooley, J. Repka, and the author. In fact, these results are closely related with Horn's conjecture about the supports of these distributions; this conjecture describes these supports in terms of honeycombs and was only recently proved by A. A. Klyachko, A. K. Knutson, and T. Tao. Some of the results in the lecture notes under review are sketched for actions of general compact Lie groups.