An Elementary Approach to Weight Multiplicities in Bivariate Irreducible Representations of Sp(2r)

By bivariate irreducible representations of ${\rm Sp}(2r)$, we mean irreducible representations with highest weights containing at most two nonzero entries, using the usual identification of dominant weights for complex symplectic Lie algebras and their corresponding Lie groups as $r$-tuples in decreasing non-negative integers. This paper has two aims. The first aim is to provide a formula for the weight mulitplicities of said representations, which is easily computable. The second aim is to present these weight multiplicities using elementary means. The formula for these weight multiplicities is derived using basic multiliear algebra and combinatorial arguments through explicit descriptions of weight vectors.