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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
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥4,000,000 (Direct Cost: ¥4,000,000)
Fiscal Year 2004: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2003: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2002: ¥1,500,000 (Direct Cost: ¥1,500,000)
Keywordsfinite group / representation theory / modular representation theory / block / abelian defect group / Broue's conjecture / Morita equivalence / irreducible character / デイド予想 / 共役類
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.

Report

(4 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (29 results)

All 2005 2004 2003 2002 Other

All Journal Article (17 results) Publications (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
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Blocks of central p-group extensions2005

    • Author(s)
      Shigeo Koshitani
    • Journal Title

      Proceedings of the American Mathematical Society 133

      Pages: 21-26

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The indecomposability of a certain bimodule given by the Brauer construction2005

    • Author(s)
      Shigeo Koshitani
    • Journal Title

      Journal of Algebra 285

      Pages: 726-729

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On sharp characters of rank 2 with rational values2005

    • Author(s)
      Sohei Nozawa
    • Journal Title

      Journal of Algebra (印刷中)

    • Related Report
      2004 Annual Research Report
  • [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
    • Related Report
      2004 Final Research Report Summary
  • [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
    • Related Report
      2004 Final Research Report Summary
  • [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
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [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
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On a 5-design related to an extremal doubly even self-dual code of length 722004

    • Author(s)
      Masaaki Kitazume
    • Journal Title

      Journal of Combinatorial Theory (A) 107

      Pages: 143-146

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Image directe superieure et unipotence2004

    • Author(s)
      Shigeki Matsuda
    • Journal Title

      Journal fur die reine und angewandte Mathematik 569

      Pages: 47-54

    • Related Report
      2004 Annual Research Report
  • [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
    • Related Report
      2004 Final Research Report Summary
  • [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
    • Related Report
      2004 Final Research Report Summary
  • [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
    • Related Report
      2004 Final Research Report Summary
  • [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
    • Related Report
      2004 Final Research Report Summary
  • [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
    • Related Report
      2004 Final Research Report Summary
  • [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
    • Related Report
      2004 Final Research Report Summary
  • [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
    • Related Report
      2004 Final Research Report Summary
  • [Publications] Shigeo Koshitani: "Conjectures of Donovan and Puig for principal 3-blocks with abelian defect groups"Communications in Algebra. 31・5. 2229-2243 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Shigeo Koshitani: "Blocks of central p-group extensions"Proceedings of the American Mathematical Society. (発表予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Shigeo Koshitani: "中心的p-群拡大である有限群のブロックについて"京都大学数理解析研究所講究録. (発表予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Shigeo Koshitani: "Examples of Broue's abelian defect group conjecture in representation theory of finite groups"Proceedings of the 36th Symposium on Ring Theory and Representation Theory. (発表予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Masaaki Kitazume: "3-state Potts model, Moonshine vertex operator algebra, and 3A-elements of the Monster group"International Mathematical Research Notice. 23. 1269-1303 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Koji Nishida: "Hilbert Coefficients and Buchsbaumness of associated graded rings"Journal of Pure and Applied Algebra. 181・1. 61-74 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Shigeo Koshitani: "Broue's conjecture holds for principal 3-blocks with elementary abelian defect group of order 9"Journal of Algebra. 248. 575-604 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Shigeo Koshitani: "Broue's conjecture for non-principal 3-blocks of finite groups"Journal of Pure and Applied Algebra. 173. 177-211 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Shigeo Koshitani: "Conjectures of Donovan and Puig for principal 3-blocks with abelian defect groups"Communications in Algebra. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] Masaaki Kitazume: "Ternary code construction of unimodular lattices and self-dual codes over Z6"Journal of Algebraic Combinatorics. 16. 209-223 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Masaaki Kitazume: "The radical subgroups of the Fischer simple group"Journal of Algebra. 255. 22-58 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Shigeki Matsuda: "Katz correspondence for quasi-unipotent overconvergent isocrystals"Composito Mathematica. 134. 1-34 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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