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 $S{p}_{2m}=S{p}_{2m}(C)$ is equivalent to branching from $S{p}_{2n}$ to $S{p}_{2n_1}\times \cdots \times S{p}_{2n_k}$ where $n,n_1,\dots ,n_k$ are positive integers such that $n=n_1+\cdots +n_k$ and the $n_j$'s depend on $m$ as well as the representations in the tensor product. Using this result and a work of J. Lepowsky, we obtain a skew Pieri rule for $S{p}_{2m}$, i.e., a description of the irreducible decomposition of the tensor product of an irreducible representation of the symplectic group $S{p}_{2m}$ with a fundamental representation.

References in corpus (2)

• Reciprocity Algebras and Branching for Classical Symmetric Pairs
• Pieri rules for classical groups and equinumeration between generalized oscillating tableaux and semistandard tableaux

Cited by in corpus (1)

• Matrix spherical analysis on nilmanifolds