Orbit measures, random matrix theory and interlaced determinantal processes

TL;DR

This paper explores the relationship between compact group representations and invariant Hermitian matrix ensembles, demonstrating that many such ensembles are determinantal through semiclassical approximations involving tensor and restriction multiplicities.

Contribution

It introduces a novel connection between representation theory and invariant matrix ensembles, extending classical Gaussian and Laguerre ensembles and showing their determinantal structure.

Findings

Many invariant ensembles are determinantal

Connections established between representation theory and matrix ensembles

Semiclassical approximations describe measures on orbits

Abstract

A connection between representation of compact groups and some invariant ensembles of Hermitian matrices is described. We focus on two types of invariant ensembles which extend the Gaussian and the Laguerre Unitary ensembles. We study them using projections and convolutions of invariant probability measures on adjoint orbits of a compact Lie group. These measures are described by semiclassical approximation involving tensor and restriction mulltiplicities. We show that a large class of them are determinantal.