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Reseach on modular representations and standard modules of association schemes

Research Project

Project/Area Number 17K05165
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

Hanaki Akihide  信州大学, 学術研究院理学系, 教授 (50262647)

Project Period (FY) 2017-04-01 – 2022-03-31
Project Status Completed (Fiscal Year 2021)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywordsアソシエーションスキーム / モジュラー表現 / 標準加群 / アダマール行列 / Terwilliger 代数 / シュアー環 / 分解体 / 一般四元数群 / タウリガー代数 / association scheme / Hadamard matrix / Frame number / adjacency algebra / 表現 / 中心化環 / 代数的組合せ論 / アソシエーション・スキーム / 代数学 / 組合せ論 / 加群
Outline of Final Research Achievements

Let p be a prime, F a field of characteristic p. We consider representations over F. We proved the standard module of a p-scheme is indecomposable. We decided the indecomposable decomposition of an association scheme obtained by repeated wreath products of class 2 association schemes. For a schurian scheme, we proved that the standard module is indecomposable if and only if the scheme is a p-scheme. This is not true for non-schurian schemes. We considered the double centralizer of the adjacency algebra in the full matrix algebra, and proved that the double centralizer is equal to the adjacency algebra if the rank of the scheme is at most 3. We also gave examples such that the double centralizer is not equal to the adjacency algebra of a scheme of rank greater than 3.

Academic Significance and Societal Importance of the Research Achievements

アソシエーションスキームは代数的組合せ論の中心的な研究対象である。この分野では考える対象のサイズが大きくなるとその構造は極めて難しくなる。そこでやや情報を落として、その隣接代数とその表現を考える「表現論」が有効になる。このときに「どの程度情報が保たれるか」を考えることは重要である。考える体が複素数体であるときにはこれまでにも多くの研究があるが、正標数の体上の研究は少ない。本研究では正標数の体上の表現を研究し、特にその直既約分解に関する多くの結果を得た。得られた結果はこれまでにはない、新しい内容のものである。表現に研究において、その直既約分解は基本的であり、今後の更なる研究と応用に期待したい。

Report

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

    (27 results)

All 2022 2021 2020 2019 2018 2017 Other

All Int'l Joint Research (8 results) Journal Article (12 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 12 results,  Open Access: 2 results,  Acknowledgement Compliant: 2 results) Presentation (4 results) (of which Int'l Joint Research: 1 results,  Invited: 1 results) Remarks (3 results)

  • [Int'l Joint Research] Steklov Mathematical Institute(ロシア連邦)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] University ofLethbridge(カナダ)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] AnhuiUniversity(中国)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] IPM(イラン)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] University ofLethbridge(カナダ)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] AnhuiUniversity(中国)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] IPM(イラン)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] University of Western Australia(オーストラリア)

    • Related Report
      2018 Research-status Report
  • [Journal Article] On a Huge Family of Non-Schurian Schur Rings2022

    • Author(s)
      Hanaki Akihide、Hirai Takuto、Ponomarenko Ilia
    • Journal Title

      The Electronic Journal of Combinatorics

      Volume: 29 Issue: 2 Pages: 1-7

    • DOI

      10.37236/10696

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] A construction of pairs of non-commutative rank 8 association schemes from non-symmetric rank 3 association schemes2021

    • Author(s)
      Hanaki Akihide、Yoshikawa Masayoshi
    • Journal Title

      Algebraic Combinatorics

      Volume: 4 Issue: 3 Pages: 533-540

    • DOI

      10.5802/alco.167

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Modular Terwilliger Algebras of Association Schemes2021

    • Author(s)
      Hanaki Akihide
    • Journal Title

      Graphs and Combinatorics

      Volume: 37 Issue: 5 Pages: 1521-1529

    • DOI

      10.1007/s00373-021-02363-0

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Classification of skew‐Hadamard matrices of order 32 and association schemes of order 312020

    • Author(s)
      Hanaki Akihide、Kharaghani Hadi、Mohammadian Ali、Tayfeh‐Rezaie Behruz
    • Journal Title

      Journal of Combinatorial Designs

      Volume: 28 Issue: 6 Pages: 421-427

    • DOI

      10.1002/jcd.21706

    • Related Report
      2020 Research-status Report 2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Double centralizers of association schemes2019

    • Author(s)
      Hanaki Akihide
    • Journal Title

      Linear Algebra and its Applications

      Volume: 564 Pages: 1-14

    • DOI

      10.1016/j.laa.2018.11.018

    • Related Report
      2019 Research-status Report 2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Frame numbers, splitting fields, and integral adjacency algebras of commutative association schemes2019

    • Author(s)
      Hanaki Akihide
    • Journal Title

      Archiv der Mathematik

      Volume: 114 Issue: 3 Pages: 259-263

    • DOI

      10.1007/s00013-019-01389-4

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Tamaschke’s Results on Schur Rings and a Generalization of Association Schemes2018

    • Author(s)
      Hanaki A.
    • Journal Title

      Journal of Mathematical Sciences

      Volume: 234 Issue: 2 Pages: 248-255

    • DOI

      10.1007/s10958-018-4000-0

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Indecomposability of modular standard modules of schurian association schemes and p-schemes2018

    • Author(s)
      Hanaki Akihide
    • Journal Title

      Archiv der Mathematik

      Volume: 印刷中 Issue: 1 Pages: 43-46

    • DOI

      10.1007/s00013-018-1166-0

    • Related Report
      2018 Research-status Report 2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Modular representation theory of BIB designs2017

    • Author(s)
      A. Hanaki, Y. Miyazaki, O. Shimabukuro
    • Journal Title

      Linear Algebra Appl.

      Volume: 514 Pages: 174-197

    • DOI

      10.1016/j.laa.2016.10.030

    • NAID

      120007100285

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Simplicity of p-blocks of modular adjacency algebras of association schemes2017

    • Author(s)
      A. Hanaki
    • Journal Title

      J. Algebra

      Volume: 474 Pages: 126-133

    • DOI

      10.1016/j.jalgebra.2016.11.012

    • NAID

      120007100303

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Indecomposable decompositions of modular standard modules for two families of association schemes2017

    • Author(s)
      Hanaki Akihide、Shimabukuro Osamu
    • Journal Title

      Journal of Algebraic Combinatorics

      Volume: 46 Issue: 2 Pages: 445-453

    • DOI

      10.1007/s10801-017-0758-2

    • NAID

      120007100302

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Tamaschke's results on Schur rings and a generalization of association schemes2017

    • Author(s)
      Hanaki Akihide
    • Journal Title

      Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)

      Volume: 455 Pages: 197-208

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Presentation] Frame numbers, splitting fields, and integral adjacency algebras of commutative association schemes2019

    • Author(s)
      花木章秀
    • Organizer
      第 36 回代数的組合せ論シンポジウム
    • Related Report
      2019 Research-status Report
  • [Presentation] Double centralizers of association schemes2018

    • Author(s)
      花木章秀
    • Organizer
      第 35 回代数的組合せ論シンポジウム
    • Related Report
      2018 Research-status Report
  • [Presentation] Double centralizers of association schemes2018

    • Author(s)
      花木章秀
    • Organizer
      The 90th KPPY Combinatorics Seminar
    • Related Report
      2018 Research-status Report
  • [Presentation] Modular standard modules of association schemes2017

    • Author(s)
      Hanaki Akihide
    • Organizer
      Workshop on Group Theory and Algebraic Combinatorics, Novosibirsk State University, Russia
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Remarks] Classification of association schemes

    • URL

      http://math.shinshu-u.ac.jp/~hanaki/as/

    • Related Report
      2021 Annual Research Report 2020 Research-status Report 2019 Research-status Report 2018 Research-status Report 2017 Research-status Report
  • [Remarks] Classification of small association schemes,

    • URL

      https://zenodo.org/record/3627821

    • Related Report
      2019 Research-status Report
  • [Remarks] GAP Package AssociationSchemes

    • URL

      http://www.jesselansdown.com/AssociationSchemes/

    • Related Report
      2018 Research-status Report

URL: 

Published: 2017-04-28   Modified: 2023-01-30  

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