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Let K be a complex reductive algebraic group and V a representation of K. LetS denote the ring of polynomials on V. Assume that the action of K on S ismultiplicity free. If V_{\lambda} is an irreducible representation of K, letS_{\lambda} denote the corresponding isotypic component of S. Write S_{\lambda}S_{\mu} for the subspace of S spanned by products of S_{\lambda} and S_{\mu}.If V_{\nu} occurs as an irreducible constituent of the tensor product ofV_{\lambda} and V_{\mu}, is it true that S_{\nu} is contained in S_{\lambda}S_{\mu}? We investigate this question for representations arising in thecontext of Hermitian symmetric pairs. We show that the answer is yes in somecases and, using an earlier result of Ruitenburg, that in the remainingclassical cases, the answer is yes provided that a conjecture of Stanley on themultiplication of Jack polynomials is true. We also show how the conjectureconnects multiplication in the ring S to the usual Littlewood-Richardson rule.

Multiplication of Polynomials on Hermitian Symmetric spaces and Littlewood–Richardson Coefficients