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Maximal Weight Composition Factors for Weyl Modules
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- Journal:
- Canadian Mathematical Bulletin / Volume 60 / Issue 4 / 01 December 2017
- Published online by Cambridge University Press:
- 20 November 2018, pp. 762-773
- Print publication:
- 01 December 2017
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- Article
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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 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.
