Symmetric Tensors and Combinatorics for Finite-Dimensional Representations of Symplectic Lie Algebras

Abstract

First, we develop a result using multilinear algebra to prove, in an elementary way, a useful identity between representations of $\mathfrak{sp}(4, \mathbb{C})$, which involves writing any irreducible representation as a formal combination of tensor products of symmetric powers of the standard representation. Once establishing this identity, we employ a combinatorial argument along with this identity to explicitly determine the weight multiplicities of any irreducible representation of $\mathfrak{sp}(4, \mathbb{C})$. While there is already a closed formula for these multiplicities, our approach is more basic and more easily accessible. After determining these multiplicities, we use them to create a method for computing the $L$- and $\varepsilon$-factors of ${\rm Sp}(4)$. Finally, we provide an approach to producing any irreducible representation of any rank $m$ symplectic Lie algebra as a formal combination of tensor products of symmetric powers of the standard representation, including a general formula given an appropriately large highest weight.