Vol. 11, No. 4, 2017

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

Ivan Dimitrov and Mike Roth

Vol. 11 (2017), No. 4, 767–815

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.

Homogeneous variety, Littlewood–Richardson coefficient, Borel–Weil–Bott theorem, PRV component.
Mathematical Subject Classification 2010
Primary: 14F25
Secondary: 17B10
Received: 28 January 2013
Revised: 11 December 2016
Accepted: 13 January 2017
Published: 18 June 2017
Ivan Dimitrov
Department of Mathematics and Statistics
Queen’s University
Jeffery Hall
Kingston ON K7L 3N6
Mike Roth
Department of Mathematics and Statistics
Queen’s University
Jeffery Hall
Kingston ON K7L 3N6