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

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

2010-02-18

arxiv.org/abs/1002.3457v6

Mathematics

Representation Theory

Scientific paper

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.

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