#### 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
##### Abstract

Let $X=G∕B$ and let ${L}_{1}$ and ${L}_{2}$ be two line bundles on $X$. Consider the cup-product map

${H}^{{d}_{1}}\left(X,{L}_{1}\right)\otimes {H}^{{d}_{2}}\left(X,{L}_{2}\right)\underset{}{\overset{\cup }{\to }}{H}^{d}\left(X,L\right),$

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 $\left({\lambda }_{1},{\lambda }_{2},\lambda \right)$ corresponding to the representations $\left({V}_{1},{V}_{2},V\right)$ 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.
Primary: 14F25
Secondary: 17B10