Abstract. We describe the inequalities on the possible eigenvalues of products of unitary matrices in terms of quantum Schubert calculus. Related problems are the existence of flat connections on the punctured two-sphere with prescribed holonomies, and the decomposition of fusion product of representations of SU (n), in the large level limit.In the second part of the paper we investigate how various aspects of the problem (symmetry, factorization) relate to properties of the Gromov-Witten invariants.

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