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

TL;DR

This paper introduces an elementary, easily computable formula for weight multiplicities in bivariate irreducible representations of Sp(2r), using basic algebra and combinatorics, simplifying previous approaches.

Contribution

It provides a new elementary method and explicit formula for calculating weight multiplicities in specific symplectic group representations, making the process more accessible.

Findings

Derived a simple, explicit formula for weight multiplicities.

Used elementary multilinear algebra and combinatorics.

Facilitated easier computation of representation weights.

Abstract

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.