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

TL;DR

This paper establishes a reciprocity law linking tensor product decompositions of symplectic group representations to branching rules, and derives a skew Pieri rule for these groups using skew duality and prior work.

Contribution

It introduces a new reciprocity law for symplectic group representations and derives a skew Pieri rule for tensor products involving fundamental representations.

Findings

Decomposition of tensor products is equivalent to branching from $Sp_{2n}$ to subgroups.

Derived a skew Pieri rule for $Sp_{2m}$.

Connected tensor product decomposition with branching rules.

Abstract

We use the theory of skew duality to show that decomposing the tensor product of $k$ irreducible representations of the symplectic group $Sp_{2m} = Sp_{2m}(C)$ is equivalent to branching from $Sp_{2n}$ to $Sp_{2n_1}\times\cdots\times Sp_{2n_k}$ where $n, n_1,\ldots, n_k$ 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 $Sp_{2m}$, i.e., a description of the irreducible decomposition of the tensor product of an irreducible representation of the symplectic group $Sp_{2m}$ with a fundamental representation.