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Cohomology theory of finite groups from the viewpoint of representation theory

Research Project

Project/Area Number 16K05054
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

Hida Akihiko  埼玉大学, 教育学部, 教授 (50272274)

Project Period (FY) 2016-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2019: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
Fiscal Year 2018: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2017: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
Fiscal Year 2016: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywords有限群の表現 / モジュラー表現 / 有限群のコホモロジー / 分類空間 / 両側Burnside環 / 有限群の表現論 / 有限群 / コホモロジー / モジュラー表現論 / 有限群の分類空間 / 両側 Burnside 環 / stable splitting / 有限群の既約指標 / Bunrside 環 / Burnside 環
Outline of Final Research Achievements

In this research, we study the relation among the following three objects on finite groups, stable splitting of classifying space, p-local structure for a prime p, modular representations. We consider rank two p-groups, in particular, we study extra special p-groups. We determine the stable irreducible summands of stable splitting and the cohomology of the summand. We use the representations of double Burnside algebra and the theory of biset functors .

Academic Significance and Societal Importance of the Research Achievements

有限群の分類空間の安定分解は両側Burnside 多元環の表現論と結びつき、可換p-群については1980年代に既に研究されている。最小の非可換群である位数が p の 3 乗のextra special p-群についても様々に研究され豊かな状況が見出されてきた。本研究では、モジュラー表現と両側集合関手の応用により、コホモロジー環の両側 Burnside 環上の加群としての既約組成因子を決定した。この結果は分類空間の安定分解についての結果として解釈することができ、ホモトピー論においても重要な情報を与えている。

Report

(5 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (11 results)

All 2020 2019 2018 2016

All Journal Article (5 results) (of which Open Access: 3 results,  Peer Reviewed: 2 results) Presentation (6 results) (of which Invited: 3 results)

  • [Journal Article] Character degree products and class length products of some finite groups2020

    • Author(s)
      A. Hida, M. Kiyota
    • Journal Title

      数理解析研究所講究録

      Volume: 2148 Pages: 1-9

    • NAID

      40022258168

    • Related Report
      2019 Annual Research Report
    • Open Access
  • [Journal Article] Character degrees and class lengths in p-blocks of some finite groups2019

    • Author(s)
      A. Hida, M. Kiyota
    • Journal Title

      数理解析研究所講究録

      Volume: 2134 Pages: 1-7

    • NAID

      120006888092

    • Related Report
      2019 Annual Research Report
    • Open Access
  • [Journal Article] Harada's conjecture on character degrees and class sizes --- symmetric and alternating groups ---2018

    • Author(s)
      飛田明彦
    • Journal Title

      数理解析研究所講究録

      Volume: 2086 Pages: 144-153

    • Related Report
      2018 Research-status Report
    • Open Access
  • [Journal Article] The splitting of cohomology of p-groups with rank 22018

    • Author(s)
      Akihiko Hida and Nobuaki Yagita
    • Journal Title

      Forum Math.

      Volume: 30 Issue: 1 Pages: 191-212

    • DOI

      10.1515/forum-2016-0241

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Representations of the double Burnside algebra and cohomology of the extraspecial p-group II2016

    • Author(s)
      A. Hida, N. Yagita
    • Journal Title

      J. Algebra

      Volume: 451 Pages: 461-493

    • DOI

      10.1016/j.jalgebra.2015.12.003

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Presentation] 有限群の既約指標の次数の積と共役類の長さの積の関係について2020

    • Author(s)
      飛田明彦
    • Organizer
      日本数学会年会
    • Related Report
      2019 Annual Research Report
  • [Presentation] Lower defect groups and vertices of simple modules2020

    • Author(s)
      飛田明彦
    • Organizer
      日本数学会年会
    • Related Report
      2019 Annual Research Report
  • [Presentation] Character degrees and class lengths in p-blocks of some finite groups2019

    • Author(s)
      飛田明彦 清田正夫
    • Organizer
      RIMS 共同研究(公開型)「有限群のコホモロジー論とその周辺」
    • Related Report
      2018 Research-status Report
  • [Presentation] Character degree products and class length products of some finite groups2018

    • Author(s)
      飛田明彦 清田正夫
    • Organizer
      RIMS共同研究 (公開型) 「代数的組合せ論と関連する群と代数の研究」
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] Harada's conjecture on character degrees and class sizes --- symmetric and alternating groups---2018

    • Author(s)
      飛田明彦
    • Organizer
      RIMS 共同研究(公開型)代数的組合せ論および有限群・頂点作用素代数とその表現の研究
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Biset functors and cohomology of finite groups2016

    • Author(s)
      飛田明彦
    • Organizer
      第61回代数学シンポジウム
    • Place of Presentation
      佐賀大学(佐賀県 佐賀市)
    • Related Report
      2016 Research-status Report
    • Invited

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Published: 2016-04-21   Modified: 2021-02-19  

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