Symmetric tensors and combinatorics for finite-dimensional representations of symplectic Lie algebras

First, we develop a result using multilinear algebra to prove, in an elementary way, a useful identity between representations of [special characters omitted], 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 [special characters omitted]. 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 ϵ-factors of 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