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On the Characters of the Syolw \(2\)-Subgroup of \(F_4(2^n)\) and Decomposition Numbers

  • In this thesis, we deal with the finite group of Lie type \(F_4(2^n)\). The aim is to find information on the \(l\)-decomposition numbers of \(F_4(2^n)\) on unipotent blocks for \(l\neq2\) and \(n\in \mathbb{N}\) arbitrary and on the irreducible characters of the Sylow \(2\)-subgroup of \(F_4(2^n)\). S. M. Goodwin, T. Le, K. Magaard and A. Paolini have found a parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), a Sylow \(2\)-subgroup of \(F_4(q)\), of \(F_4(p^n)\), \(p\) a prime, for the case \(p\neq2\). We managed to adapt their methods for the parametrization of the irreducible characters of the Sylow \(2\)-subgroup for the case \(p=2\) for the group \(F_4(q)\), \(q=p^n\). This gives a nearly complete parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), namely of all irreducible characters of \(U\) arising from so-called abelian cores. The general strategy we have applied to obtain information about the \(l\)-decomposition numbers on unipotent blocks is to induce characters of the unipotent subgroup \(U\) of \(F_4(q)\) and Harish-Chandra induce projective characters of proper Levi subgroups of \(F_4(q)\) to obtain projective characters of \(F_4(q)\). Via Brauer reciprocity, the multiplicities of the ordinary irreducible unipotent characters in these projective characters give us information on the \(l\)-decomposition numbers of the unipotent characters of \(F_4(q)\). Sadly, the projective characters of \(F_4(q)\) we obtained were not sufficient to give the shape of the entire decomposition matrix.

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Metadaten
Author:Ulrike Faltings
URN (permanent link):urn:nbn:de:hbz:386-kluedo-52051
Advisor:Gunter Malle
Document Type:Doctoral Thesis
Language of publication:English
Publication Date:2018/12/04
Year of Publication:2018
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2018/03/23
Date of the Publication (Server):2018/04/13
Number of page:V,207
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):Creative Commons 4.0 - Namensnennung, nicht kommerziell, keine Bearbeitung (CC BY-NC-ND 4.0)