Let alpha = (alpha_1,...,alpha_n), beta = (beta_1,..., beta_n), and gamma = ( gamma_1,...,gamma_n) be three sequences of n real numbers. In this talk we will describe inequalities for the alpha's, beta's and gamma's that were conjectured by Alfred Horn (and recently proved true by Klyacho, Knutson, and Tao) as necessary and sufficient for the existence of Hermitian matrices A, B, and C with C = A + B, such that alpha is the sequence of eigenvalues of A, beta is the sequence of eigenvalues of B, and gamma is the sequence of eigenvalues of C. We shall also show the connection between the Horn conjecture and similar relations between partitions that satisfy the Littlewood-Richardson rule.

By that mean we refer relations between the problem on the eigenvalues of sum of hermitian matrices with the irreducible homogeneous representations of the full linear group and the representations of the symmetric group.

Other problems connected with the Horn conditions would also be referred.