Cup products of line bundles on homogeneous varieties and generalized PRV components of multiplicity one

Details Cup products of line bundles on homogeneous varieties and generalized PRV components of multiplicity one Cup products of line bundles on homogeneous varieties and generalized PRV components of multiplicity one

2009-09-14

arxiv.org/abs/0909.2280v1

Mathematics

Algebraic Geometry

61 pages, 2 figures, uses PStricks

Scientific paper

Let X=G/B be a complete flag variety, and L' and L" two line bundles on X. Consider the cup product map H^{d'}(X,L') x H^{d"}(X, L") --> H^{d}(X,L), where L=L' x L" and d=d'+d". We answer two natural questions about the map above: When is it a nonzero map of irreducible G-representations? Conversely, given generic irreducible representations V' and V" of G, which irreducible components of V' x V" may appear in the right hand side of the map above? We also give bounds on the multiplicities appearing in a tensor product, and relate these considerations to the boundary of the Littlewood-Richardson cone.

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