Extremal rays of the equivariant Littlewood-Richardson cone

TL;DR

This paper introduces an inductive method to identify extremal rays of the equivariant Littlewood-Richardson cone, linking it to Hermitian eigenvalue problems and extending classical algorithms.

Contribution

It develops a new inductive procedure for the equivariant Littlewood-Richardson cone, solving a rational version of a previously posed problem and extending classical algorithms.

Findings

Provided an inductive procedure for extremal rays

Connected the cone to Hermitian eigenvalue problems

Analyzed special rays and Hilbert basis properties

Abstract

We give an inductive procedure for finding the extremal rays of the equivariant Littlewood-Richardson cone, which is closely related to the solution space to S. Friedland's majorized Hermitian eigenvalue problem. In so doing, we solve the "rational version" of a problem posed by C. Robichaux, H. Yadav, and A. Yong. Our procedure is a natural extension of P. Belkale's algorithm for the classical Littlewood-Richardson cone. The main tools for accommodating the equivariant setting are certain foundational results of D. Anderson, E. Richmond, and A. Yong. We also study two families of special rays of the cone and make observations about the Hilbert basis of the associated lattice semigroup.