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

where $L = L_{1} \otimes L_{2}$ and $d = d_{1} + d_{2}$ . We answer two natural questions about the map above: When is it a nonzero homomorphism of representations of $G$ ? Conversely, given generic irreducible representations $V_{1}$ and $V_{2}$ , which irreducible components of $V_{1} \otimes V_{2}$ 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 $V_{1} \otimes V_{2}$ of stable multiplicity one. Furthermore, the highest weights $(λ_{1}, λ_{2}, λ)$ corresponding to the representations $(V_{1}, V_{2}, V)$ 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.

Keywords

Homogeneous variety, Littlewood–Richardson coefficient, Borel–Weil–Bott theorem, PRV component.

Mathematical Subject Classification 2010

Primary: 14F25

Secondary: 17B10

Milestones

Received: 28 January 2013

Revised: 11 December 2016

Accepted: 13 January 2017

Published: 18 June 2017

Authors

Ivan Dimitrov
Department of Mathematics and Statistics Queen’s University Jeffery Hall Kingston ON K7L 3N6 Canada

Mike Roth
Department of Mathematics and Statistics Queen’s University Jeffery Hall Kingston ON K7L 3N6 Canada

Vol. 11, No. 4, 2017

Ivan Dimitrov and Mike Roth

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors