A non-recursive criterion for weights of a highest weight module for an affine Lie algebra

TL;DR

This paper presents a new non-recursive criterion to determine if a certain weight belongs to a highest weight module of an affine Lie algebra, simplifying previous recursive methods.

Contribution

It introduces a novel non-recursive criterion based on coefficients modulo an integral lattice, enabling efficient computation of weight inclusion.

Findings

The criterion is expressed in terms of coefficients modulo an integral lattice.

The set needed for the criterion can be computed efficiently.

The method simplifies the process of identifying weights in highest weight modules.

Abstract

Let $\Lambda$ be a dominant integral weight of level $k$ for the affine Lie algebra $\mathfrak g$ and let $\alpha$ be a non-negative integral combination of simple roots. We address the question of whether the weight $\eta=\Lambda-\alpha$ lies in the set $P(\Lambda)$ of weights in the irreducible highest-weight module with highest weight $\Lambda$. We give a non-recursive criterion in terms of the coefficients of $\alpha$ modulo an integral lattice $kM$, where $M$ is the lattice parameterizing the abelian normal subgroup $T$ of the Weyl group. The criterion requires the preliminary computation of a set no larger than the fundamental region for $kM$, and we show how this set can be efficiently calculated.