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2004 Fiscal Year Final Research Report Summary

For the solution of Broue's conjecture in representation theory of finite groups

Research Project

Project/Area Number 14540009
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionChiba University

Principal Investigator

KOSHITANI Shigeo  Chiba University, Faculty of Science, Professor, 理学部, 教授 (30125926)

Co-Investigator(Kenkyū-buntansha) NOZAWA Sohei  Chiba University, Faculty of Science, Professor, 理学部, 教授 (20092083)
KITAZUME Masaaki  Chiba University, Faculty of Science, Professor, 理学部, 教授 (60204898)
MATSUDA Shigeki  Chiba University, Faculty of Science, Associated Professor, 理学部, 助教授 (90272301)
NISHIDA Koji  Chiba University, School of Science, Associated Professor, 大学院・自然科学研究院, 助教授 (60228187)
OTSUBO Noriyuki  Chiba University, School of Science, Assistant Professor, 大学院・自然科学研究院, 助手 (60332566)
Project Period (FY) 2002 – 2004
Keywordsfinite group / representation theory / modular representation theory / block / abelian defect group / Broue's conjecture / Morita equivalence / irreducible character
Research Abstract

The head investigator Koshitani has gotten the following 3 results in this research.
First, we get a result that if our block is the principal 3-block with elementary abelian defect group of order 9 of an arbitrary finite group, then Broue's conjecture holds anytime. This may be one of the most important results on the conjecture. This was published in the "Journal of Algebra" (Elsevier.).
Because of the above result, it should be natural to consider Broue's conjectre as the next step for non-principal 3-block of finite groups with the same elementary abelian defect group of order 9. We did it for other concrete finite groups, say O'Nan's simple group and the Higman-Sims simple group which are sporadic, namely we proved Broue's conjecture is still true for non-principal 3'blocks of these finite simple groups with the same defect group. The result was published in the "Journal of Pure and Applied Algebra" (Elsevier).
There are conjectures of Donovan and Puig, both of which are closely related to Broue's abelian defect group conjecture. As the third result in this research, we proved that these two conjectures hold for all principal 3-blocks of finite groups with the same Sylow 3-subgroups which are elementary abelian of order 9. This result was published in a journal "Communications inAlgebra (Dekker)".
The head investigator Koshitani announced these 3 results in University of Durham, UK 2002,Torun University, Poland 2003,Banff Science Institute, Canada 2003 and also University of Oxford, UK 2004,as invited talks.
We have had another important results due to the other investigators Kitazume and Matsuda, for instance.

  • Research Products

    (12 results)

All 2005 2004 2003 2002

All Journal Article (12 results)

  • [Journal Article] Broue's conjecture holds for principal 3-blocks with elementary abelian defect group of order 92005

    • Author(s)
      Shigeo Koshitani
    • Journal Title

      Journal of Algebra 248

      Pages: 575-604

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Broue's abelian defect group conjecture for the Held group and the specific Suzuki group2004

    • Author(s)
      Shigeo Koshitani
    • Journal Title

      Journal of Algebra 279

      Pages: 638-666

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Image directe superieure et unipotence2004

    • Author(s)
      Shigeki Matsuda
    • Journal Title

      Journal fuer die reine und angewandte Mathematik 569

      Pages: 47-54

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Broue's abelian defect group conjecture for the Held group and the sporadic Suzuki group2004

    • Author(s)
      Shigeo Koshitani
    • Journal Title

      Journal of Algebra 279

      Pages: 638-666

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Image direct superieure et unipotent2004

    • Author(s)
      Shigeki Matsuda
    • Journal Title

      Journal fuer die reine und angewandte Mathematik 569

      Pages: 47-54

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Conjectures of Donovan and puig for principal 3-blocks with abelian defect gropus2003

    • Author(s)
      Shigeo Koshitani
    • Journal Title

      Communications in Algebra 31

      Pages: 2229-2243

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Conjectures of Donovan and Puig for principal 3-blocks with abelian defect groups2003

    • Author(s)
      Shigeo Koshitani
    • Journal Title

      Communications in Algebra 31

      Pages: 2229-2243

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Broue's conjecture dor non-principal 3-blocks of finite gropus2002

    • Author(s)
      Shigeo Koshitani
    • Journal Title

      Journal of Pure and Applied Algebra 173

      Pages: 177-211

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] The radical subgroups of the Fischet simple group2002

    • Author(s)
      Masaaki Kitazume
    • Journal Title

      Journal of Algebra 255

      Pages: 22-58

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Broue's conjecture holds for principal 3'blocks with elementary abelian defect group of order 92002

    • Author(s)
      Shigeo Koshitani
    • Journal Title

      Journal of Algebra 248

      Pages: 575-604

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Broue's conjecture for non-principal 3-blocks of finite groups2002

    • Author(s)
      Shigeo Koshitani
    • Journal Title

      Journal of Pure and Applied Algebra 173

      Pages: 177-211

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The radical subgroups of the Fischer simple group2002

    • Author(s)
      Masaaki Kitazume
    • Journal Title

      Journal of Algebra 255

      Pages: 22-58

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2006-07-11  

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