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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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 42, No. 1, pp. 53–62, 2008
Original Russian Text Copyright © by D. I. Panyushev and O. S. Yakimova
Both authors are supported by RFBR grants nos. 05-01-00988 and 06-01-72550. The second author gratefully acknowledges the support of the Alexander von Humboldt-Stiftung.
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Panyushev, D.I., Yakimova, O.S. The PRV-formula for tensor product decompositions and its applications. Funct Anal Its Appl 42, 45–52 (2008). https://doi.org/10.1007/s10688-008-0005-7
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DOI: https://doi.org/10.1007/s10688-008-0005-7