  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 V_{\lambda} is an irreducible representation of K, let
S_{\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 of
V_{\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 the
context of Hermitian symmetric pairs. We show 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. We also show how the conjecture
connects multiplication in the ring S to the usual Littlewood-Richardson rule.
