Abstract
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 ⊂ 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.
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