Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians

TL;DR

This paper connects eigenvalue problems of Hermitian matrices with Schubert calculus on Grassmannians, providing a new interpretation and saturation theorem for Friedland's generalized eigenvalue problem using equivariant cohomology.

Contribution

It introduces a Schubert calculus perspective on Friedland's eigenvalue problem and proves a saturation theorem within this framework, linking matrix eigenvalues to Grassmannian cohomology.

Findings

Established a Schubert calculus interpretation of Friedland's eigenvalue problem

Proved a saturation theorem for the generalized eigenvalue problem

Connected eigenvalue problems with equivariant cohomology of Grassmannians

Abstract

The saturation theorem of [Knutson-Tao '99] concerns the nonvanishing of Littlewood-Richardson coefficients. In combination with work of [Klyachko '98], it implies [Horn '62]'s conjecture about eigenvalues of sums of Hermitian matrices. This eigenvalue problem has a generalization [Friedland '00] to majorized sums of Hermitian matrices. We further illustrate the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians. Our main result gives a Schubert calculus interpretation of Friedland's problem, via equivariant cohomology of Grassmannians. In particular, we prove a saturation theorem for this setting. Our arguments employ the aformentioned work together with [Thomas-Yong '12].