Newell-Littlewood numbers III: eigencones and GIT-semigroups

TL;DR

This paper extends the understanding of Newell-Littlewood numbers by characterizing the eigencone, providing linear inequalities, and connecting these to Hermitian matrix eigenvalues, generalizing classical results for Littlewood-Richardson coefficients.

Contribution

It establishes an eigenvalue interpretation, minimal inequalities, and a factorization for the saturated NL-cone, advancing the theory of tensor product multiplicities for classical groups.

Findings

Eigenvalue interpretation of NL-cone

Minimal linear inequalities for NL-cone

Factorization of NL-numbers on the boundary

Abstract

The Newell-Littlewood numbers are tensor product multiplicities of Weyl modules for the classical groups in the stable range. Littlewood-Richardson coefficients form a special case. Klyachko connected eigenvalues of sums of Hermitian matrices to the saturated LR-cone and established defining linear inequalities. We prove analogues for the saturated NL-cone: an eigenvalue interpretation; a minimal list of defining linear inequalities; a description by Extended Horn inequalities, as conjectured in part II of this series; and a factorization of NL-numbers, on the boundary.