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Research on the structure of generalized Burnside rings and its application

Research Project

Project/Area Number 16K05052
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionMuroran Institute of Technology

Principal Investigator

Takegahara Yugen  室蘭工業大学, 大学院工学研究科, 教授 (10211351)

Project Period (FY) 2016-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Keywordsバーンサイド環 / 斜バーンサイド環 / 単項バーサイド環 / 単数群 / テンソル誘導写像 / 乗法的誘導写像 / テンソル誘導 / 乗法的誘導 / 抽象バーンサイド環 / Mバーンサイド環 / ゴースト環 / 線形指標 / モノイド / 単項バーンサイド環 / F-バーンサイド環 / オブストラクション環 / Hyperoctahedral群 / 対称群 / 指標環 / 置換指標 / 置換表現 / 有限群 / 量子2重構成
Outline of Final Research Achievements

Regarding the structure of the Burnside ring of a finite group, we obtained some results about the unit group. In addition, regarding the generalization of the Burnside ring, we have collectively constructed the examples that have been known so far and unifiedly showed the known properties. The construction made it possible to carry out research on multiplicative induction maps and unit groups in a unified manner. In particular, with regard to the multiplicative induction map, we succeeded in giving sufficient conditions for the existence of a generalization of tensor induction maps between Burnside rings.

Academic Significance and Societal Importance of the Research Achievements

有限群のバーンサイド環とその一般化の研究は過去50年に渡って続けられている。特に、今世紀になってからは、多くの研究が発表されてきた。そこで、個別に進められてきた、様々なバーンサイド環の研究を統一的に行い、その上で、さらにバーンサイド環の一般化の例を与えることで、有限群の研究に応用することができないかという問いが生まれてきた。本研究はその問題に取り組む端緒を開いた。得られた諸結果は、今後、さらに取り組むべき問題を与えている。

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

    (8 results)

All 2019 2018 2017

All Journal Article (4 results) (of which Peer Reviewed: 4 results) Presentation (4 results) (of which Invited: 1 results)

  • [Journal Article] The number of subgroups of a finite group (II)2019

    • Author(s)
      Yugen Takegahara
    • Journal Title

      Communications in Algebra

      Volume: 47 Issue: 5 Pages: 1964-1972

    • DOI

      10.1080/00927872.2018.1527918

    • NAID

      120006648056

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] p-adic estimates of the number of permutation representations2019

    • Author(s)
      Yugen Takegahara
    • Journal Title

      Advances in Mathematics

      Volume: 349 Pages: 367-425

    • DOI

      10.1016/j.aim.2019.04.008

    • NAID

      120006711440

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Axiomatic theory of Burnside rings. (I)2018

    • Author(s)
      T. Yoshida, F. Oda, and Y. Takegahara
    • Journal Title

      Journal Algebra

      Volume: 505 Pages: 339-382

    • DOI

      10.1016/j.jalgebra.2018.03.012

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Lefschetz invariants and Young characters for representations of the hyperoctahedral groups2018

    • Author(s)
      F. Oda, Y. Takegahara, and T. Yoshida
    • Journal Title

      Journal Algebra

      Volume: 512 Pages: 1-19

    • DOI

      10.1016/j.jalgebra.2018.07.001

    • NAID

      120006529948

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Presentation] モノミアル・バーンサイド環の乗法的性質について2019

    • Author(s)
      竹ヶ原裕元
    • Organizer
      有限群のコホモロジー論とその周辺, 京都大学数理解析研究所
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] モノミアル・バーンサイド環の積構造について2019

    • Author(s)
      竹ヶ原裕元
    • Organizer
      有限群のコホモロジー論とその周辺 RIMS 共同研究(公開型)
    • Related Report
      2018 Research-status Report
  • [Presentation] Axiomatic theory of Burnside rings I2018

    • Author(s)
      小田文仁, 竹ヶ原裕元, 吉田知行
    • Organizer
      2018年日本数学会年会, 東京大学
    • Related Report
      2018 Research-status Report
  • [Presentation] Lefschetz invariants and Young characters for representations of the Coxeter groups of type B2017

    • Author(s)
      小田文仁, 竹ヶ原裕元, 吉田知行
    • Organizer
      日本数学会2017年度秋季総合分科会
    • Related Report
      2017 Research-status Report

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

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