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The centralizer of a subgroup in a group algebra

Part of: Representation theory of groups

Published online by Cambridge University Press:  20 November 2012

Susanne Danz ,
Harald Ellers  and
John Murray
Susanne Danz
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles', Oxford OX1 3LB, UK
Harald Ellers
Affiliation:
Departament of Mathematics, Allegheny College, Meadville, PA 16335, USA ([email protected])
John Murray
Affiliation:
Department of Mathematics, National University of Ireland, Maynooth, County Kildare, Ireland, ([email protected])
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Abstract

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Let F be an algebraically closed field, G be a finite group and H be a subgroup of G. We answer several questions about the centralizer algebra FGH. Among these, we provide examples to show that

  • • the centre Z(FGH) can be larger than the F-algebra generated by Z(FG) and Z(FH),

  • • FGH can have primitive central idempotents that are not of the form ef, where e and f are primitive central idempotents of FG and FH respectively,

  • • it is not always true that the simple FGH-modules are the same as the non-zero FGH-modules HomFH(S, T ↓ H), where S and T are simple FH and FG-modules, respectively.

Keywords

centralizer algebrasHecke algebrassimple modulescentral idempotentsblocks

MSC classification

Secondary: 20C05: Group rings of finite groups and their modules 20C08: Hecke algebras and their representations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 1 , February 2013 , pp. 49 - 56
Copyright
Copyright © Edinburgh Mathematical Society 2012

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