Oppositeness in buildings and simple modules for finite groups of Lie type

Details Oppositeness in buildings and simple modules for finite groups of Lie type Oppositeness in buildings and simple modules for finite groups of Lie type

2011-02-26

arxiv.org/abs/1102.5404v1

Mathematics

Group Theory

14 pages

Scientific paper

In the building of a finite group of Lie type we consider the incidence relations defined by oppositeness of flags. Such a relation gives rise to a homomorphism of permutation modules (in the defining characteristic) whose image is a simple module for the group. The $p$-rank of the incidence relation is then the dimension of this simple module. We give some general reductions towards the determination of the character of the simple module. Its highest weight is identified and the problem is reduced to the case of a prime field. The reduced problem can be approached through the representation theory of algebraic groups and the methods are illustrated for some examples.

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