The PRV-Formula for Tensor Product Decompositions and Its Applications

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.