Eigenvalues of majorized Hermitian matrices and Schubert calculus

Recent work of Klyachko, Totaro, Knutson, Tao, Belkale and Woodward has given sharp answers to the problem of characterizing the possible eigenvalues of Hermitian matrices A, B, and C, with C = A + B. We answer the more general problem when C is required only to be less than A + B, i.e., A + B - C is positive semidefinite. The proof requires, in addition to the results already mentioned, a little Schubert calculus. We also discuss a related problem of characterizing the finitely generated torsion modules that can appear in an exact sequence A -> C -> B, and the (open) problem for longer exact sequences.