The symplectic and algebraic geometry of Horn's problem

Abstract

Horn's problem was the following: given two Hermitian matrices with known spectra, what might be the eigenvalue spectrum of the sum? This linear algebra problem is exactly of the sort to be approached with the methods of modern Hamiltonian geometry (which were unavailable to Horn). The theorem linking symplectic quotients and geometric invariant theory lets one also bring algebraic geometry and representation theory into play. This expository note is intended to elucidate these connections for linear algebraists, in the hope of making it possible to recognize what sort of problems are likely to fall to the same techniques that were used in proving Horn's conjecture.