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Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants. (English) Zbl 1041.14025
Summary: The set of conjugacy classes appearing in a product of conjugacy classes in a compact, \(1\)-connected Lie group \(K\) can be identified with a convex polytope in the Weyl alcove. In this paper we identify linear inequalities defining this polytope. Each inequality corresponds to a non-vanishing Gromov-Witten invariant for a generalized flag variety \(G/P\), where \(G\) is the complexification of \(K\) and \(P\) is a maximal parabolic subgroup. This generalizes the results for \(SU(n)\) of S. Agnihotri and C. Woodward [Math. Res. Lett. 5, No. 6, 817–836 (1998; Zbl 1004.14013)] and P. Belkale [Compos. Math. 129, No. 1, 67–86 (2001; Zbl 1042.14031)] on the eigenvalues of a product of unitary matrices and quantum cohomology of Grassmannians.

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14M15 Grassmannians, Schubert varieties, flag manifolds
05E99 Algebraic combinatorics
14L30 Group actions on varieties or schemes (quotients)
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