Geodesic
The PRV-Formula for Tensor Product Decompositions and Its Applications
Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 1, pp. 53-62

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Let $G$ be a semisimple algebraic group, $V$ a simple finite-dimensional self-dual $G$-module, and $W$ an arbitrary simple finite-dimensional $G$-module. Using the triple multiplicity formula due to Parthasarathy, Ranga Rao, and Varadarajan, we describe the multiplicities of $W$ in the symmetric and exterior squares of $V$ in terms of the action of a maximum-length element of the Weyl group on some subspace in $V^T$, where $T\subset G$ is a maximal torus. By way of application, we consider the cases in which $V$ is the adjoint, little adjoint, or, more generally, a small $G$-module. We also obtain a general upper bound for triple multiplicities in terms of Kostant's partition function.
Keywords: semisimple Lie algebra, highest weight, triple multiplicity, partition function. 
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     author = {D. I. Panyushev and O. S. Yakimova},
     title = {The {PRV-Formula} for {Tensor} {Product} {Decompositions} and {Its} {Applications}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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D. I. Panyushev; O. S. Yakimova. The PRV-Formula for Tensor Product Decompositions and Its Applications. Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 1, pp. 53-62. http://geodesic.mathdoc.fr/item/FAA_2008_42_1_a4/