Tensor Products of Highest Weight Representations and Skew-Symmetric Matrix Equations A+B+C=0

Abstract: The question of characterizing the eigenvalues for the sum of two Hermitian matrices, was solved in 1999, after almost a century of efforts. The saturation conjecture for GL_C(n) was proven by Knutson and Tao, filling in the last gap in Horn’s conjecture. Under certain conditions, this problem is equivalent to decomposing the tensor product of two finite dimensional irreducible highest weight representations of GL_C(n). In the first part of this thesis we use the methods of moment maps and coadjoint orbits to find equivalence between the eigenvalue problem for skew-symmetric matrices and the decomposition of tensor products of irreducible highest weight representations of SO_C(2k). We characterize the eigenvalues in the cases k = 2,3, where we can take advantage of Lie algebra isomorphisms. In the second part, we consider irreducible, infinite dimensional, unitary highest weight representations of GL_C(n + 1) as representations on spaces of vector valued polynomials, and we find irreducible factors in the tensor product of two such representations.