M. Schaps | Searcharxiv

arXiv2011

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

Let $Λ$ be a dominant integral weight of level $k$ for the affine Lie algebra $\mathfrak g$ and let $α$ be a non-negative integral combination of simple roots. We address the question of whether the weight $η=Λ-α$ lies in the set $P(Λ)$ of weights in the irreducible highest-weight module with highest weight $Λ$. We give a non-recursive criterion in terms of the coefficients of $α$ 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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arXiv2009

Crossover Morita equivalences of spin representations of the symmetric and alternating groups

We prove the existence of Morita equivalences between the spin blocks at the extremal points of strings in the block-reduced crystal graph. When the parities of the core partitions are not preserved, these equivalences require crossovers, with a block of the symmtric group Morita equivalent to a block of the alternating group and vice versa. The result permits in improvement of the bound for Donovan's Conjecture given by Kessar.

math.RT↗