On the distribution of the spectrum of the sum of two hermitian or real symmetric matrices

Let A, B and C = U₁ A U₁^* + U₂ B U₂^* be hermitian (or real symmetric) matrices, where U₁ and U₂ are unitary (respectively orthogonal). Our goal is to develop as concrete as possible expression to the probability distribution of the spectrum of C where U_i are drawn from the unitary (respectively orthogonal) group with respect to the Haar measure. While we do this for the unitary group, using representation theory of U_n, we can only accomplish the same for the real symmetric case where B has rank 1, by performing explicit calculations. Here is what we do in the unitary case. For a given n by n matrix A over the field of complex numbers we study the operator E_A^N = ∫ u A u^{*⊗ N} d u, where the integration is taken over the unitarian group with respect to the Haar measure. When A has nonnegative spectrum, we show that E_A^N, as N tends to infinity, is concentrated around some simple S_N × U_n submodule of (Cⁿ)^{⊗ N} determined only by the spectra of A. Along the proof we reprove a generalization of Heckman to the convexity theorem of Horn. The Schur-Weyl duality and the technique of Bernstein polynomials approximations are our tools. Using this we compute the distribution, induced by a sum of two hermitian orbits on the set of hermitian orbits in terms of asymptotic Littlewood-Richardson coefficients times an asymptotic version of the Weyl dimensional formula. More precisely, translation of the lattice permutation and semistandard Young tableaux rules to a set of linear inequalities, enable us to express the density of the distribution above as multiplication of volumes of two concrete polytopes.