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2017 Cup products of line bundles on homogeneous varieties and generalized PRV components of multiplicity one
Ivan Dimitrov, Mike Roth
Algebra Number Theory 11(4): 767-815 (2017). DOI: 10.2140/ant.2017.11.767

Abstract

Let X = GB and let L1 and L2 be two line bundles on X. Consider the cup-product map

Hd1 (X,L1) Hd2 (X,L2) Hd(X,L),

where L = L1 L2 and d = d1 + d2. We answer two natural questions about the map above: When is it a nonzero homomorphism of representations of G? Conversely, given generic irreducible representations V1 and V2, which irreducible components of V1 V2 may appear in the right hand side of the equation above? For the first question we find a combinatorial condition expressed in terms of inversion sets of Weyl group elements. The answer to the second question is especially elegant: the representations V appearing in the right hand side of the equation above are exactly the generalized PRV components of V1 V2 of stable multiplicity one. Furthermore, the highest weights (λ1,λ2,λ) corresponding to the representations (V1,V2,V) fill up the generic faces of the Littlewood–Richardson cone of G of codimension equal to the rank of G. In particular, we conclude that the corresponding Littlewood–Richardson coefficients equal one.

Citation

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Ivan Dimitrov. Mike Roth. "Cup products of line bundles on homogeneous varieties and generalized PRV components of multiplicity one." Algebra Number Theory 11 (4) 767 - 815, 2017. https://doi.org/10.2140/ant.2017.11.767

Information

Received: 28 January 2013; Revised: 11 December 2016; Accepted: 13 January 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1367.14017
MathSciNet: MR3665637
Digital Object Identifier: 10.2140/ant.2017.11.767

Subjects:
Primary: 14F25
Secondary: 17B10

Rights: Copyright © 2017 Mathematical Sciences Publishers

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