A reciprocity law and the skew Pieri rule for the symplectic group

Abstract

We use the theory of skew duality to show that decomposing the tensor product of k irreducible representations of the symplectic group Sp2m=Sp2m(ℂ) is equivalent to branching from Sp2n to Sp2n1×⋯×Sp2nk, where n,n1,…,nk are positive integers such that n=n1+⋯+nk and the njs depend on m as well as the representations in the tensor product. Using this result and a work of Lepowsky, we obtain a skew Pieri rule for Sp2m, i.e., a description of the irreducible decomposition of the tensor product of an irreducible representation of the symplectic group Sp2m with a fundamental representation.