Charaktertafeln parabolischer Untergruppen der Steinbergschen Trialitätsgruppen und Anwendungen auf deren Darstellungstheorie
The first part of this thesis deals with the character tables of the maximal parabolic subgroups of Steinberg's triality groups. First, an algorithm is presented, based on the Steinberg presentation, for generic computations in finite groups of Lie type and connected reductive algebraic groups. Using this algorithm, the conjugacy classes of the maximal parabolic subgroups of the triality groups are determined. Then, by Clifford theory, the generic character table of a maximal parabolic subgroup and a large part of the generic character table of another maximal parabolic subgroup are computed. In 1990, M. Geck has determined the decomposition numbers of Steinberg's triality groups in non-defining characteristic, leaving some ambiguities in the principal block. In the second part of this thesis, the character tables and the modular representation theory of the maximal parabolic subgroups are used to prove new results on these decomposition numbers. By an analysis of restrictions of suitable modules from the triality groups to the maximal parabolic subgroups using Green correspondence, two of the so far unknown decomposition numbers are determined, and for several others the known bounds are improved.
info:eu-repo/classification/ddc/510, Mathematik, Charaktertafeln, parabolische Untergruppen, Steinberg, Trialitätsgruppen, Darstellungstheorie, Zerlegungszahlen
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