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From spin representations of the symmetric groups to Hirota equations

Research Project

Project/Area Number 17K05180
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionOkayama University (2022)
Kumamoto University (2017-2021)

Principal Investigator

YAMADA HiroFumi  岡山大学, 自然科学研究科, 特命教授 (40192794)

Project Period (FY) 2017-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
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)
Keywordsシューア函数 / 対称群 / ヴィラソロ代数 / Q函数 / 無限次元リー環 / ソリトン方程式 / KdV方程式 / KdV方程式系 / 広田双線型表示 / プリュッカー関係式 / ヤング図形 / 広田方程式 / 代数学 / 関数方程式論
Outline of Final Research Achievements

I am studying combinatorial aspects of Hirota bilinear equations for Kdv and modified KdV hierarchies. Mikio Sato wrote a short notes on these objects in Japanese in 1980, and tabulated his experimental results. Only recently I have realized what Sato was thinking and saying. Schur functions and Schur's Q functions are essentially related to the subject.
And also modular representations of the symmetric groups at p=2 have a deep connection with thesehierarchies. I am also investigating certain representation of the Virasoro algebra, which, I think, is related to KdV. Three papers on the reduced Fock representation of the Virasoro algebra appeared which are written jointly with Kazuya Aokage and Eriko Shinkawa.
And one more paper with Naoki Chigira will appear, I hope, which is aninteresting combinatorics of partitions.

Academic Significance and Societal Importance of the Research Achievements

非線型偏微分方程式の一つであるKdV方程式は,ソリトンと呼ばれる特別な解をもつ.自然現象の解明におおいに寄与するが,一方数学的にもその代数性等際立った性質を具備している.私は,佐藤幹夫による無限次元グラスマン多様体の理論を土台にしつつ,組合せ論的な側面に着目して研究を進めている.非線型現象を表す微分方程式が対称群という小さな代数系の組合せ論で制御されている,ということを明らかにしつつあるという意味で
学術的な意義は小さくないと信ずる.

Report

(7 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • Research Products

    (14 results)

All 2022 2021 2020 2019 2018 2017

All Journal Article (6 results) (of which Peer Reviewed: 6 results,  Open Access: 4 results) Presentation (8 results) (of which Int'l Joint Research: 5 results,  Invited: 6 results)

  • [Journal Article] Virasoro action on the Q-functions2021

    • Author(s)
      Kazuya Aokage, Eriko Shinkawa and Hiro-Fumi Yamada
    • Journal Title

      Symmetry, Integrability and Geometry

      Volume: 17

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Pfaffian identities and Virasoro operators2020

    • Author(s)
      K.Aokage, E.Shinkawa and H.-F.Yamada
    • Journal Title

      Letters in Mathematical Physics

      Volume: 110 Issue: 6 Pages: 1381-1389

    • DOI

      10.1007/s11005-020-01265-1

    • Related Report
      2020 Research-status Report 2019 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Arithmetic identities for class regular partitions2018

    • Author(s)
      Hiroshi Mizukawa and HiroFumi Yamada
    • Journal Title

      Hokkaido Mathematical Journal

      Volume: 47 Pages: 205-221

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Reduced Fock representation of the Virasoro algebra2018

    • Author(s)
      HiroFumi Yamada
    • Journal Title

      代数的組合せ論シンポジウム2018報告集

      Volume: - Pages: 38-45

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Arithmetic identities for class regular partitions2018

    • Author(s)
      Hiroshi MIZUKAWA and HiroFumi YAMADA
    • Journal Title

      Hokkaido Mathematical Journal

      Volume: 47

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Products of parts in class regular partitions2017

    • Author(s)
      Masanori ANDO and HiroFumi YAMADA
    • Journal Title

      Hiroshima Mathematical Journal

      Volume: 47

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] Differential Plucker relations2022

    • Author(s)
      HiroFumi Yamada
    • Organizer
      モンゴル国立大学数学コロキウム
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] KdV方程式系の広田表示2021

    • Author(s)
      西山雄太,山田裕史
    • Organizer
      日本数学会九州支部例会
    • Related Report
      2021 Research-status Report
  • [Presentation] Reduced Fock representation of the Virasoro algebra2020

    • Author(s)
      山田裕史
    • Organizer
      日本数学会九州支部例会
    • Related Report
      2019 Research-status Report
  • [Presentation] Reduced Fock representation of the Virasoro algebra2019

    • Author(s)
      HiroFumi Yamada
    • Organizer
      Advances and Perspectives in Representation Theory , Qingdao
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Reduced Fock representation of the Virasoro algebra2018

    • Author(s)
      HiroFumi Yamada
    • Organizer
      Symposium of Algebraic Combinatorics
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] ヴィラソロ代数の被約フォック表現2018

    • Author(s)
      山田裕史
    • Organizer
      福井表現論小研究集会
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] On Sato's "On Hirota's Bilinear Equations"2018

    • Author(s)
      HiroFumi YAMADA
    • Organizer
      Algebraic and Enumerative Combinatorics in Okayama
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On Sato's "On Hirota's Bilinear Equations"2017

    • Author(s)
      HiroFumi YAMADA
    • Organizer
      日中表現論研究会(上海)
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2017-04-28   Modified: 2024-01-30  

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