A symplectic proof of the Horn inequalities

TL;DR

This paper provides a symplectic proof of the Horn inequalities, connecting eigenvalue sums of Hermitian matrices with symplectic geometry, tropical calculus, and combinatorics, and offers a tropical perspective on the Duistermaat-Heckman measure.

Contribution

It introduces a novel symplectic approach to prove Horn inequalities and describes the Duistermaat-Heckman measure tropically, combining multiple mathematical techniques.

Findings

Symplectic proof of Horn inequalities.

Tropical description of the Duistermaat-Heckman measure.

Integration of tropical calculus, combinatorics, and symplectic geometry.

Abstract

In this paper, we give a symplectic proof of the Horn inequalities on eigenvalues of a sum of two Hermitian matrices with given spectra. Our method is a combination of tropical calculus for matrix eigenvalues, combinatorics of planar networks, and estimates for the Liouville volume. As a corollary, we give a tropical description of the Duistermaat-Heckman measure on the Horn polytope.