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Multiplication of polynomials on Hermitian symmetric spaces and Littlewood-Richardson coefficients. (English) Zbl 1206.14071

Let \(K\) be a complex reductive algebraic group and let \(V\) be a linear representation of \(K\). Assume that the natural action of \(G\) on \(S=\mathbb C[V]\), the ring of polynomials on \(V\), is multiplicity free. If \(\lambda\) is the isomorphism class of a simple \(G\)-module, fix a choice of a simple \(G\)-module \(V_\lambda\) in \(\lambda\). Let \(S_\lambda\) be the \(\lambda\)-isotypic component of \(S\). The authors consider the following question. If \(\nu\) is the class of a simple submodule of \(V_\lambda\otimes V_\mu\), is it true that \(S_\nu\) is contained in the subspace of \(S\) spanned by the products \(ab\) where \(a\in S_\lambda\), \(b\in S_\mu\)? The authors investigate this question for representations arising in the context of Hermitian symmetric pairs. They show that the answer is positive in some cases and that in the remaining classical cases it is positive as well provided that a conjecture of Stanley on the multiplication of Jack polynomials is true.

MSC:

14L30 Group actions on varieties or schemes (quotients)
22E46 Semisimple Lie groups and their representations
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