Quantum generalization of the Horn conjecture

Abstract

(Recall that the Lie algebra of the special unitary group SU(n) is isomorphic to the real vector space of traceless Hermitian matrices as representations of SU(n) and hence the terminology eigenvalue for SU(n).) In 1962, Horn [17] gave a conjectural solution to a problem equivalent to the above additive eigenvalue problem for SU(n), by a recursively determined system of inequalities. In [19], Klyachko gave a solution to the additive eigenvalue problem for SU(n) in terms of a certain system of inequalities. To write down this system, we need to know which structure constants in the cohomology of Gr(r, n) (written in the Schubert basis) 0 2 be a positive integer. Characterize the possible eigenvalues (a1, a2,..., as) of matrices A^\A^2 ..., A? e SU(n) which satisfy A A& A^ = 1.