Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-20T13:23:34.005Z Has data issue: false hasContentIssue false
  • English
  • Français

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 (hellers@allegheny.edu)
John Murray
Affiliation:
Department of Mathematics, National University of Ireland, Maynooth, County Kildare, Ireland, (john.murray@nuim.ie)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1.Alperin, J. L., Local representation theory, Cambridge Studies in Advanced Mathematics, Volume 11 (Cambridge University Press, 1986).CrossRefGoogle Scholar
2.Alperin, J. L., On the center of a Hecke algebra, J. Alg. 319(2) (2008), 777–778.CrossRefGoogle Scholar
3.Bosma, W., Cannon, J. and Playoust, C., The Magma algebra system, I, The user language, J. Symb. Computat. 24 (1997), 235–265.CrossRefGoogle Scholar
4.Ellers, H., The defect groups of a clique, p-solvable groups, and Alperin's conjecture, J. Reine Angew. Math. 468 (1995), 1–48.Google Scholar
5.Ellers, H., Searching for more general weight conjectures, using the symmetric group as an example, J. Alg. 225 (2000), 602–629.CrossRefGoogle Scholar
6.Ellers, H. and Murray, J., Block theory, branching rules, and centralizer algebras, J. Alg. 276(1) (2004), 236–258.CrossRefGoogle Scholar
7.Ellers, H. and Murray, J., Blocks of centralizer algebras and affine Hecke algebras, submitted.Google Scholar
8. GAP Group, GAP—groups, algorithms, programming—a system for computational discrete algebra, Version 4.4.10 (2007) (available at www.gap-system.org).Google Scholar
9.James, G. and Kerber, A., The representation theory of the symmetric group, Encyclopedia of Mathematics and Its Applications, Volume 16 (Addison-Wesley, Reading, MA, 1981).Google Scholar
10.Kleshchev, A., Linear and projective representations of symmetric groups (Cambridge University Press, 2005).CrossRefGoogle Scholar
11.Külshammer, B., Group-theoretical descriptions of ring-theoretical invariants of group algebras, in Representation theory of finite groups and finite-dimensional algebras, Progress in Mathematics, Volume 95, pp. 425–442 (Birkhäuser, 1991).CrossRefGoogle Scholar
12.Robinson, G. R., Some remarks on Hecke algebras, J. Alg. 163(3) (1994), 806–812.CrossRefGoogle Scholar