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Maximal Weight Composition Factors for Weyl Modules

 Printed: Dec 2017
  • Jens Carsten Jantzen,
    Institut for Matematik, Aarhus Universitet , Ny Munkegade 118, DK-8000 Aarhus C , Denmark
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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 to any dominant weight $\lambda$ for $R$ two $G_k$--modules with highest weight $\lambda$, the Weyl module $V (\lambda)_k$ and its simple quotient $L (\lambda)_k$. Let $\lambda$ and $\mu$ be dominant weights with $\mu \lt \lambda$ 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 classification of the possible pairs $(\lambda, \mu)$, and another one that relies only on the classification of root systems.
Keywords: algebraic groups, represention theory algebraic groups, represention theory
MSC Classifications: 20G05, 20C20 show english descriptions Representation theory
Modular representations and characters
20G05 - Representation theory
20C20 - Modular representations and characters

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