Oppositeness in Buildings and Simple Modules for Finite Groups of Lie Type
نویسنده
چکیده
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.
منابع مشابه
Simplicity of Modules given by Oppositeness Relations in Spherical Buildings
Oppositeness graphs of a spherical building are generalizations of the classical Kneser graphs. Recently Brouwer [1] has shown that the square of each eigenvalue of the adjacency matrix of an oppositeness graph is a power of q, for buildings of finite groups of Lie type defined over Fq. Here we show that the incidence modules are simple. The essential part of the proof is a result of Carter and...
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