Moment polytopes in real symplectic geometry II : applications to singular value inequalities

TL;DR

This paper explores convex cones linked to isotropic representations of symmetric spaces, using cohomological conditions to describe inequalities, with a focus on the singular Horn cone related to singular value inequalities for rectangular matrices.

Contribution

It introduces a cohomological framework to characterize convex cones associated with isotropic representations, extending classical eigenvalue inequalities to singular values.

Findings

Characterization of convex cones via cohomological conditions

Description of singular Horn cone inequalities

Extension of classical eigenvalue inequalities to singular values

Abstract

In this work, we study some convex cones associated to isotropic representations of symmetric spaces. We explain the inequalities that describe them by means of cohomological conditions. In particular, we study the singular Horn cone which is the counterpart of the classical Horn cone, where the eigenvalues of Hermitian square matrices are replaced by the singular values of rectangular matrices.