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The Inductive Blockwise Alperin Weight Condition for the Finite Groups \( SL_3(q) \) \( (3 \nmid (q-1)) \), \( G_2(q) \) and \( ^3D_4(q) \)

  • The central topic of this thesis is Alperin's weight conjecture, a problem concerning the representation theory of finite groups. This conjecture, which was first proposed by J. L. Alperin in 1986, asserts that for any finite group the number of its irreducible Brauer characters coincides with the number of conjugacy classes of its weights. The blockwise version of Alperin's conjecture partitions this problem into a question concerning the number of irreducible Brauer characters and weights belonging to the blocks of finite groups. A proof for this conjecture has not (yet) been found. However, the problem has been reduced to a question on non-abelian finite (quasi-) simple groups in the sense that there is a set of conditions, the so-called inductive blockwise Alperin weight condition, whose verification for all non-abelian finite simple groups implies the blockwise Alperin weight conjecture. Now the objective is to prove this condition for all non-abelian finite simple groups, all of which are known via the classification of finite simple groups. In this thesis we establish the inductive blockwise Alperin weight condition for three infinite series of finite groups of Lie type: the special linear groups \(SL_3(q)\) in the case \(q>2\) and \(q \not\equiv 1 \bmod 3\), the Chevalley groups \(G_2(q)\) for \(q \geqslant 5\), and Steinberg's triality groups \(^3D_4(q)\).

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Metadaten
Author:Elisabeth Schulte
URN (permanent link):urn:nbn:de:hbz:386-kluedo-42250
Advisor:Gunter Malle
Document Type:Doctoral Thesis
Language of publication:English
Publication Date:2015/11/06
Year of Publication:2015
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2015/10/16
Date of the Publication (Server):2015/11/09
Number of page:IX, 226
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):20-XX GROUP THEORY AND GENERALIZATIONS / 20Cxx Representation theory of groups [See also 19A22 (for representation rings and Burnside rings)] / 20C15 Ordinary representations and characters
20-XX GROUP THEORY AND GENERALIZATIONS / 20Cxx Representation theory of groups [See also 19A22 (for representation rings and Burnside rings)] / 20C20 Modular representations and characters
20-XX GROUP THEORY AND GENERALIZATIONS / 20Cxx Representation theory of groups [See also 19A22 (for representation rings and Burnside rings)] / 20C33 Representations of finite groups of Lie type
Licence (German):Standard gemäß KLUEDO-Leitlinien vom 30.07.2015