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Modules with trivial source, modular monomial representations and a modular version of Brauer's induction theorem

Modules with trivial source, modular monomial representations and a modular version of Brauer's... LetG be a finite group andK an algebraically closed field of characteristicp ≠ 0. It is shown that for any indecomposableKG-moduleM with a trivial “source” in the sense ofJ. A. Green there exist at most 1-dimensionalKV-modulesS 1(V) andS 2(V) withV ranging over all subgroups ofG with a normal Sylow-p-subgroupV p and an elementary factor groupV/V p such that $$\mathop \oplus \limits_V S^1 (V)^{V \to G} \oplus M \cong \mathop \oplus \limits_V S_2 (V)^{V \to G} $$ , i.e.M is essentially induced from 1-dimensional representations of such subgroups. Moreover it is shown that one can associate to any direct sum of suchM certain numerical invariants (characters), which distinguishM within this class of modules up to isomorphism. The work is based onR. Brauer's induction theorem andJ. A. Green's transfer theorem. It is closely related to some earlier work ofS.B. Conlon. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

Modules with trivial source, modular monomial representations and a modular version of Brauer's induction theorem

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References (12)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02992950
Publisher site
See Article on Publisher Site

Abstract

LetG be a finite group andK an algebraically closed field of characteristicp ≠ 0. It is shown that for any indecomposableKG-moduleM with a trivial “source” in the sense ofJ. A. Green there exist at most 1-dimensionalKV-modulesS 1(V) andS 2(V) withV ranging over all subgroups ofG with a normal Sylow-p-subgroupV p and an elementary factor groupV/V p such that $$\mathop \oplus \limits_V S^1 (V)^{V \to G} \oplus M \cong \mathop \oplus \limits_V S_2 (V)^{V \to G} $$ , i.e.M is essentially induced from 1-dimensional representations of such subgroups. Moreover it is shown that one can associate to any direct sum of suchM certain numerical invariants (characters), which distinguishM within this class of modules up to isomorphism. The work is based onR. Brauer's induction theorem andJ. A. Green's transfer theorem. It is closely related to some earlier work ofS.B. Conlon.

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Nov 18, 2008

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