# Extremal rays in the Hermitian eigenvalue problem

## TL;DR

This paper characterizes the extremal rays of the cones governing the Hermitian eigenvalue problem, relating them to flag variety geometry and providing explicit formulas for their construction.

## Contribution

It introduces a novel geometric approach to identify extremal rays of the eigenvalue cones using flag varieties and induction methods.

## Key findings

- Extremal rays are linked to modular intersection loci.
- Explicit formulas for extremal rays from intersection loci.
- Induction from smaller groups generates additional extremal rays.

## Abstract

The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of $n\times n$ Hermitian matrices, given the eigenvalues of the summands. The regular faces of the cones $\Gamma_n(s)$ controlling this problem have been characterized in terms of classical Schubert calculus by the work of several authors. We determine extremal rays of $\Gamma_n(s)$ (which are never regular faces) by relating them to the geometry of flag varieties: The extremal rays either arise from "modular intersection loci", or by "induction" from extremal rays of smaller groups. Explicit formulas are given for both the extremal rays coming from such intersection loci, and for the induction maps.