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Broue's conjecture in representation theory of finite groups and related topics

Research Project

Project/Area Number 15K04776
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionChiba University

Principal Investigator

Shigeo Koshitani  千葉大学, 先進科学センター, 特任教授 (30125926)

Project Period (FY) 2015-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥5,850,000 (Direct Cost: ¥4,500,000、Indirect Cost: ¥1,350,000)
Fiscal Year 2017: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2016: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2015: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywordsブルエ予想 / アルペリン重み予想 / アルペリン・マッカイ予想 / 表現論 / ブロック / 自明自己準同型加群 / 導来同値 / 箙 / カルタン不変数
Outline of Final Research Achievements

In order to solve "Broue's conjecture" we must have the Brauer indecomposability of the Scott modules. We proved it for some cases. As related topics of Broue's conjecture we have two conjectures, say "Alperin's weight conjecture" and "Alperin-McKay conjecture. For the case where the defect group is cyclic we have obtained three results. There is also an important notion "Endo-trivial modules" related to the conjectures. We proved it for dihedral defect groups case which had been an open problem. All the work was joint one with four persons in the UK and Germany. I gave invited talks e.g. in Oberwolfach and Jena in Germany 2015, in EPFL in Lausanne Switzerland 2016, and also in Banff Canada 2017.

Report

(4 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (36 results)

All 2018 2017 2016 2015 Other

All Int'l Joint Research (12 results) Journal Article (12 results) (of which Int'l Joint Research: 10 results,  Peer Reviewed: 11 results,  Open Access: 1 results,  Acknowledgement Compliant: 5 results) Presentation (9 results) (of which Int'l Joint Research: 7 results,  Invited: 9 results) Remarks (3 results)

  • [Int'l Joint Research] Wuppertal University/Kaiserslautern Technical University/Jena University(ドイツ)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] University of Milano-Bicocca(イタリア)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] City University, London/BIrmingham University(英国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] University of Illinis/City University of New York/University of Minnesota(米国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Wuppertal University/Kaiserslautern Technical University/Aachen Technical University(ドイツ)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] City University, London/Manchester University/Birmingham University(英国)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] University of Illinois/City University of New York/University of Minnesota(米国)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] University of Millano-Bicocca(イタリア)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] Jena University/Kaiserslautern Technical University/Aachen Technical University(ドイツ)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] City University London/Manchester University(英国)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] University of Illinois/City University of New York/University of Minnesota(米国)

    • Related Report
      2015 Research-status Report
  • [Int'l Joint Research] University of Milano(イタリア)

    • Related Report
      2015 Research-status Report
  • [Journal Article] Simple modules in the Auslander-Reiten quiver of principal blocks with abelian defect groups2018

    • Author(s)
      S. Koshitani, C. Lassueur
    • Journal Title

      Nagoya Mathematical Journal

      Volume: 印刷中 Pages: 1-28

    • DOI

      10.1017/nmj.2017.45

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Source algebra version of Donovan's conjecture for finite group algebras2018

    • Author(s)
      S. Koshitani
    • Journal Title

      Proceedings of the 50th Symposium on Ring Theory and Representation Theory

      Volume: 50 Pages: 105-109

    • Related Report
      2017 Annual Research Report
    • Open Access / Int'l Joint Research
  • [Journal Article] The Projective cover of the trivial representation for finite groups of Lie type in defining characteristic2017

    • Author(s)
      S. Koshitani, J. Mueller
    • Journal Title

      Algebra Colloquium

      Volume: 24 Issue: 03 Pages: 439-452

    • DOI

      10.1142/s1005386717000281

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Endo-trivial modules for finite groups with dihedral Sylow 2-subgroups2017

    • Author(s)
      S.Koshitani, C.Lassueur
    • Journal Title

      Journal of Group Theory

      Volume: 印刷中 Issue: 4 Pages: 635-660

    • DOI

      10.1515/jgth-2015-0044

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The inductive Alperin-McKay condition for 2-blocks with cyclic defect groups2016

    • Author(s)
      S. Koshitani, B. Spaeth
    • Journal Title

      Archiv der Mathematik

      Volume: 106 Issue: 2 Pages: 107-116

    • DOI

      10.1007/s00013-015-0852-4

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] The inductive Alperin-McKay and blockwise Alperin weight conditions for blocks with cyclic gefect groups2016

    • Author(s)
      S. Koshitani, B. Spaeth
    • Journal Title

      Journal of Group Theory

      Volume: 19 Issue: 5 Pages: 777-813

    • DOI

      10.1515/jgth-2016-0006

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Broue's abelian defect group conjecture and 3-decomposition numbers of sporadic simple Conway group Co12015

    • Author(s)
      S. Koshitani, J. Mueller, F. Noeske
    • Journal Title

      Journal of Pure and Applied Algebra

      Volume: 219

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Finite groups whose first Cartan invarians over a field of characteristic two are two2015

    • Author(s)
      S. Koshitani
    • Journal Title

      Journal of Algebra

      Volume: 426 Pages: 25900272-25900272

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] Clifford theory of characters in induced blocks2015

    • Author(s)
      S. Koshitani, B. Spaeth
    • Journal Title

      Proceedings of the American Mathematical Society

      Volume: 143

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Endo-trivial modules for finite groups with Klein-four Sylow 2-subgroups2015

    • Author(s)
      Shigeo Koshitani and Caroline Lassueur
    • Journal Title

      Manuscripta Mathematica

      Volume: 未定 Issue: 1-2 Pages: 265-282

    • DOI

      10.1007/s00229-015-0739-5

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] On symmetric quotients of symmetric algebras2015

    • Author(s)
      R. Kessar, S. Koshitani, M. Linckelmann
    • Journal Title

      Journal of Algebra

      Volume: 442 Pages: 423-437

    • DOI

      10.1016/j.jalgebra.2014.05.035

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] On the Brauer indecomposability of Scott module2015

    • Author(s)
      R. Kessar, S. Koshitani, M. Linckelmann
    • Journal Title

      Quarterly Journal of Mathematics

      Volume: 66 Issue: 3 Pages: 895-903

    • DOI

      10.1093/qmath/hav010

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] Simple modulesin the stable Auslaner-Reiten quivers for finite group algebras2017

    • Author(s)
      S. Koshitani
    • Organizer
      RIMS研究集会 有限群のコホモロジー論とその周辺
    • Place of Presentation
      京都大学数理解析研究所(京都府京都市)
    • Year and Date
      2017-02-16
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] 有限群の表現論における局所大域予想-最近5年間の大きな進展2017

    • Author(s)
      越谷 重夫
    • Organizer
      第62回代数学シンポジウム
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Puig equivalence conjecture for 2-blocks2017

    • Author(s)
      S. Koshitani
    • Organizer
      New Perspectives in Representation Theory of Finite Groups
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Yet Brauer indecomposability of the Scott module2017

    • Author(s)
      S. Koshitani
    • Organizer
      Darstellungstheorietage 2017
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The Z*-theorem from source algebras point of view2016

    • Author(s)
      S. Koshitani
    • Organizer
      Morita equivalence problems for blocks of finite groups
    • Place of Presentation
      スイス連邦工科大学ローザンヌ校(スイス ローザンヌ市)
    • Year and Date
      2016-09-09
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Yet small 1-PIMS2016

    • Author(s)
      S. Koshitani
    • Organizer
      Endo-p-permutation and trivial source modules in the representation theory of finite groups
    • Place of Presentation
      スイス連邦工科大学ローザンヌ校 (スイス ローザンヌ市)
    • Year and Date
      2016-09-02
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Endo-trivial modules for finite groups with dihedral Sylow 2-subgroups2016

    • Author(s)
      S. Koshitani
    • Organizer
      RIMS研究集会 有限群とその表現、頂点作用素代数、代数的組合せ論の研究
    • Place of Presentation
      京都大学数理解析研究所(京都府京都市)
    • Year and Date
      2016-01-08
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Endo-trivial modules in dihedral Sylow 2-subgorups case2015

    • Author(s)
      S. Koshitani
    • Organizer
      Blocks of Finite Groups Beyond (B.Kuelshammer's 60th Birthday Conference)
    • Place of Presentation
      Jena 大学(ドイツ・イェーナ市)
    • Year and Date
      2015-07-24
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Endo-trivial modules revisited2015

    • Author(s)
      S. Koshitani
    • Organizer
      Representations of Finite Groups
    • Place of Presentation
      Oberwolfach数学研究所(ドイツ・オーバーヴォルファッハ村)
    • Year and Date
      2015-04-09
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Remarks]

    • URL

      http://www.math.s.chiba-u.acjp/~koshitan

    • Related Report
      2017 Annual Research Report
  • [Remarks] http://www.math.s.chiba-u.ac.jp/~koshitan/

    • Related Report
      2016 Research-status Report
  • [Remarks] http://weber.math.s.chiba-u.ac.jp/~koshitan/

    • Related Report
      2015 Research-status Report

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Published: 2015-04-16   Modified: 2022-06-14  

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