Extremal rays in the Hermitian eigenvalue problem for arbitrary types

TL;DR

This paper characterizes the extremal rays of eigencones for Hermitian eigenvalue problems across all semisimple groups, generalizing previous work and providing explicit formulas and geometric interpretations.

Contribution

It introduces a comprehensive description of extremal rays for eigencones in arbitrary semisimple groups, extending prior results for SL(n) and developing new geometric and inductive methods.

Findings

Extremal rays lie on regular facets of the eigencone.

Explicit formulas for some extremal rays with geometric significance.

A new geometric process called induction from Levi subgroups to understand remaining rays.

Abstract

The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of Hermitian matrices given the eigenvalues of the summands. This is a problem about the Lie algebra of the maximal compact subgroup of $G=\operatorname{SL}(n)$ . There is a polyhedral cone (the "eigencone") determining the possible answers to the problem. These eigencones can be defined for arbitrary semisimple groups $G$, and also control the (suitably stabilized) problem of existence of non-zero invariants in tensor products of irreducible representations of $G$. We give a description of the extremal rays of the eigencones for arbitrary semisimple groups $G$ by first observing that extremal rays lie on regular facets, and then classifying extremal rays on an arbitrary regular face. Explicit formulas are given for some extremal rays, which have an explicit geometric meaning as cycle classes of interesting loci, on an arbitrary regular face, and the remaining extremal rays on that face are understood by a geometric process we introduce, and explicate numerically, called induction from Levi subgroups. Several numerical examples are given. The main results, and methods, of this paper generalize work of [Bel17] which handled the case of $G=\operatorname{SL}(n)$.