Maximal Weight Composition Factors for Weyl Modules

Jantzen, Jens Carsten

Canadian mathematical bulletin, Tome 60 (2017) no. 4, pp. 762-773

Résumé

Fix an irreducible (finite) root system $R$ and a choice of positive roots. For any algebraically closed field $k$ consider the almost simple, simply connected algebraic group ${{G}_{k}}$ over $k$ with root system $k$ . One associates with any dominant weight $\lambda $ for $R$ two ${{G}_{k}}$ -modules with highest weight $\lambda $ , the Weyl module $V{{(\lambda )}_{k}}$ and its simple quotient $V{{(\lambda )}_{k}}$ . Let $\lambda $ and $\mu $ be dominant weights with $\mu <\lambda $ such that $\mu $ is maximal with this property. Garibaldi, Guralnick, and Nakano have asked under which condition there exists $k$ such that $L{{(\mu )}_{k}}$ is a composition factor of $V{{(\lambda )}_{k}}$ , and they exhibit an example in type ${{E}_{8}}$ where this is not the case. The purpose of this paper is to to show that their example is the only one. It contains two proofs for this fact: one that uses a classiffication of the possible pairs $(\lambda ,\mu )$ , and another that relies only on the classiûcation of root systems.

Détail

DOI : 10.4153/CMB-2016-055-4

Mots-clés : 20G05, 20C20, algebraic groups, represention theory

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