A closed character formula for symmetric powers of irreducible representations

TL;DR

This paper derives a closed-form character formula for symmetric powers of irreducible representations of complex semi-simple Lie algebras, simplifying the computation of weight multiplicities through rational functions and partial fraction techniques.

Contribution

It introduces a novel closed character formula using partial fraction decomposition, connecting it to vector partition functions and providing explicit computations in special cases.

Findings

Derived a closed character formula for symmetric powers

Connected character formula to vector partition functions

Computed rational functions in specific cases

Abstract

We prove a closed character formula for the symmetric powers $S^N V(\lambda)$ of a fixed irreducible representation $V(\lambda)$ of a complex semi-simple Lie algebra $\mathfrak{g}$ by means of partial fraction decomposition. The formula involves rational functions in rank of $\mathfrak{g}$ many variables which are easier to determine than the weight multiplicities of $S^N V(\lambda)$ themselves. We compute those rational functions in some interesting cases. Furthermore, we introduce a residue-type generating function for the weight multiplicities of $S^N V(\lambda)$ and explain the connections between our character formula, vector partition functions and iterated partial fraction decomposition.