A symplectic proof of the Horn inequalities
  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. 
Anton Alekseev, Masha Podkopaeva, Andras Szenes
math.SG math.RT