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

Abstract

Abstract. Let K be a complex reductive algebraic group and V a representation of K. Let S denote the ring of polynomials on V. Assume that the action of K on S is multiplicity-free. If ƛ denotes the isomorphism class of an irreducible representation of K, let ρƛ : K → GL(Vƛ) denote the corresponding irreducible representation and Sƛ the ƛ-isotypic component of S. Write Sƛ ・ Sμ for the subspace of S spanned by products of Sƛ and Sμ. If Vν occurs as an irreducible constituent of Vƛ ⊗ Vμ, is it true that Sν ⊆ Sƛ ・ Sμ? In this paper, the authors investigate this question for representations arising in the context of Hermitian symmetric pairs. It is shown that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. It is also shown how the conjecture connects multiplication in the ring S to the usual Littlewood–Richardson rule.