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A Study on Glauberman-Watanabe Correspondence and Derived Equivalence of Blocks

Research Project

Project/Area Number 18540004
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionHokkaido University of Education

Principal Investigator

OKUYAMA Tetsuro  Hokkaido University of Education, Faculty of Education, Professor (60128733)

Co-Investigator(Kenkyū-buntansha) HASEGAWA Izumi  Hokkaido University of Education, Faculty of Education, Professor (50002473)
KITAYAMA Masashi  Hokkaido University of Education, Faculty of Education, Professor (80169888)
OKUBO Kazuyoshi  Hokkaido University of Education, Faculty of Education, Professor (80113661)
YATSUI Tomoaki  Hokkaido University of Education, Faculty of Education, Associate Professor (00261371)
IAI Shin-Ichiro  Hokkaido University of Education, Faculty of Education, Associate Professor (50333125)
佐々木 洋城  信州大学, 全学教育機構, 教授 (60142684)
Project Period (FY) 2006 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,750,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥450,000)
Fiscal Year 2007: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2006: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsRepresentation of Finite Group / Block Algebra / Derived Equivalence / Glauberman Correspondence / Complex of Modules / Cohomology / 加群の複体とコホモロジー
Research Abstract

In the research of this project, we studied "the abelian defect conjecture" which is one of main problems in representation theory of finite groups. We were mainly concerned with Glauberman-Watanabe correspondence.
1. We have constructed a two sided complex in the principal blocks of the group U(4,q^2) which gives a stable equivalence. In our investigation. we have solved conjectures on complexes of some module categories related to Alvis-Curtis-Kawanaka duality in finite reductive groups and to self-equivalences in Hecke algebras of finite Coxeter groups.
2. We studied the problem raised by Holloway-Koshitani-Kunugi which concerns with relations between blocks of SL(2,q) and its extension group by a field automorphism group. And we constructed some one sided complex for the group Sz(q) which we might expect to be a tilting complex. Blocks appearing here have non-abelian defect groups with cyclic focal subgroup.
3. We also worked on a problem concerning with quasi-perfect isometries which is a generalization of notion of perfect isometries recently studied by Narasaki-Uno. And we solved their conjecture for the group U(4,q^2) and Sz(q) over the field with defining characteristic. We studied a perfect isometriy between the blocks of the groups ^2F_4(2) and ^2F_4(q).

Report

(3 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • Research Products

    (21 results)

All 2008 2007 2006

All Journal Article (15 results) (of which Peer Reviewed: 4 results) Presentation (6 results)

  • [Journal Article] On a certain simple modules and cohomology of the symmetric group over GF(2)2008

    • Author(s)
      T. Okuyama
    • Journal Title

      RIMS Kokyuroku(数理解析研究所講究録) 1581

      Pages: 58-63

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On a certain simple modules and cohomology of the symmetric group over GF(2)2008

    • Author(s)
      T. Okuyama
    • Journal Title

      RIMS Kokyuroku 1581

      Pages: 58-63

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On a certain simple module and cohomology of the symmetric group over GF(2)2008

    • Author(s)
      T. Okuyama
    • Journal Title

      RIMS Koukyuroku(数理解析研究所講究録) 1581

      Pages: 58-63

    • Related Report
      2007 Annual Research Report
  • [Journal Article] Brauer correspondence and Green correspondence2008

    • Author(s)
      H. Sasaki
    • Journal Title

      RIMS Koukyuroku(数理解析研究所講究録) 1581

      Pages: 14-19

    • Related Report
      2007 Annual Research Report
  • [Journal Article] Parabolic geometries associated with differential equations of finite type2007

    • Author(s)
      K. Yamaguchi and T. Yatsui
    • Journal Title

      Progress in Mathematics 252

      Pages: 161-209

    • NAID

      120006459462

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Parabolic geometries associated with differential equations of finite type2007

    • Author(s)
      K. Yamaguch and T. Yatsui
    • Journal Title

      Progress in Mathematice 252

      Pages: 161-209

    • NAID

      120006459462

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Parabolic Geometries Associated with Differential Equations of Finite Type2007

    • Author(s)
      K.Yamaguchi, T.Yatsui
    • Journal Title

      Progress in Mathematics, From Geometry to Quantum Mechanics 252

      Pages: 161-209

    • NAID

      120006459462

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Cohomology algebras of blocks of finite groups and Brauer correspondence2006

    • Author(s)
      H. Kawai and H. Sasaki
    • Journal Title

      Algebras and Representation Theory 9

      Pages: 497-511

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Cohomology algebras of 2-blocks of finite groups with defect groups of rank two2006

    • Author(s)
      H. Kawai and H. Sasaki
    • Journal Title

      Journal of Algebra 306

      Pages: 301-321

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On weekly unitarily invariant norm and the \lambda-Aluthge transformations for invertible operator.2006

    • Author(s)
      K. Okubo
    • Journal Title

      Linear Algebra and its Application 419

      Pages: 48-52

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Cohomology algebras of 2- blocks of finite groups and with defect groups of Rank 22006

    • Author(s)
      H. Kawai and H. Sasaki
    • Journal Title

      Journal of Algebra 306

      Pages: 301-321

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On weekly unitarily invariant norm and the \lambda-Aluthge Transformation for invertible operator2006

    • Author(s)
      K. Okubo
    • Journal Title

      Linear Algebra and its Application 419

      Pages: 48-52

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Cohomology algebras of blocks of finite groups and Brauer correspondence2006

    • Author(s)
      H.Kawai, H.Sasaki
    • Journal Title

      Algebras and Representation Theory 9・5

      Pages: 497-511

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Cohomology algebras of 2-blocks of finite groups with defect groups of rank two2006

    • Author(s)
      H.Kawai, H.Sasaki
    • Journal Title

      Journal of Algebra 306・2

      Pages: 301-321

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On weakly unitarily invariant norm and the lambda-Aluthge transformation for invertible operator2006

    • Author(s)
      K.Okubo
    • Journal Title

      Linear Algebra and its Application 419・1

      Pages: 48-52

    • Related Report
      2006 Annual Research Report
  • [Presentation] 対称群のGF(2)上のある種の既約加群とコホモロジー2007

    • Author(s)
      奥山 哲郎
    • Organizer
      RIMS研究集会(有限群のコホモロジー論の研究)
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2007-07-29
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] On a certain simple module and cohomology of the symmetric group over GF(2)2007

    • Author(s)
      T. Okuyama
    • Organizer
      Cohomology Theory of Finite Groups and Related Topics Jul.29.2007, Research Inst. Mathematical Sci
    • Place of Presentation
      Tokyo University
    • Year and Date
      2007-07-29
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Brauer対応とGreen対応2007

    • Author(s)
      佐々木 洋城
    • Organizer
      RIMS研究集会(有限群のコホモロジー論の研究)
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2007-07-27
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Brauer correspondence and Green correspondence2007

    • Author(s)
      H. Sasaki
    • Organizer
      Cohomology Theory of Finite Groups and Related Topics Jul.27.2007, Research Inst. Mathematical Sci
    • Place of Presentation
      Tokyo University
    • Year and Date
      2007-07-27
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 有限群の表現論におけるブルエ予想をめぐって2006

    • Author(s)
      奥山 哲郎
    • Organizer
      第51回代数学シンポジウム
    • Place of Presentation
      東京大学大学院
    • Year and Date
      2006-08-05
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On Broue's Conjecture in Representation theory of Finite Groups2006

    • Author(s)
      T. Okuyama
    • Organizer
      51 st Symposium on Algebras (Math. Soc. Japan)
    • Place of Presentation
      Tokyo University
    • Year and Date
      2006-08-05
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2006-04-01   Modified: 2016-04-21  

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