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Combinatorics of finite-dimensional algebras and quantum symmetry

Research Project

Project/Area Number 22540015
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionMeijo University (2012-2014)
Kyoto University (2010-2011)

Principal Investigator

MAENO Toshiaki  名城大学, 理工学部, 准教授 (60291423)

Project Period (FY) 2010-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000)
Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2013: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2012: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords代数的組合せ論 / Hopf代数 / 鏡映群 / 旗多様体 / Lefschetz性 / 量子代数
Outline of Final Research Achievements

In this project, we have obtained the results mainly on the following two themes.
(1) We have given a description of the Schubert calculus and its generalizations in terms of noncommutative differential structures on the Weyl groups. The main topics are the K-ring of the flag variety and the homology of the affine Grassmannian. We have described them as subalgebras of the Nichols-Woronowicz algebras associated with the corresponding Weyl groups.
(2) We have obtained some results on the Lefschetz property of finite-dimensional Gorenstein algebras. We have determined the set of the Lefschetz elements of the coinvariant algebra of the finite Coxeter groups (except H_4). We have also introduced a new class of Gorenstein algebras defined by matroids and proved its Lefschetz property when the matroid corresponds to a geometric modular lattice.

Report

(6 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Annual Research Report
  • 2012 Annual Research Report
  • 2011 Annual Research Report
  • 2010 Annual Research Report
  • Research Products

    (9 results)

All 2015 2013 2012 2011

All Journal Article (5 results) (of which Peer Reviewed: 5 results) Presentation (3 results) (of which Invited: 1 results) Book (1 results)

  • [Journal Article] A Mathematical Problem for Security Analysis of Hash Functions and Pseudorandom Generators2015

    • Author(s)
      Koji Nuida, Takuro Abe, Shizuo Kaji, Toshiaki Maeno, Yasuhide Numata
    • Journal Title

      International Journal of Foundations of Computer Science

      Volume: 26 Issue: 02 Pages: 169-194

    • DOI

      10.1142/s0129054115500100

    • NAID

      120007100293

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Sperner property, matroids and finite-dimensional Gorenstein algebras.2012

    • Author(s)
      Maeno Toshiaki and Yasuhide Numata
    • Journal Title

      Contemp. Math.

      Volume: 580 Pages: 73-84

    • DOI

      10.1090/conm/580/11496

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Affine nil-Hecke algebras and braided differential structure on affine Weyl groups2012

    • Author(s)
      Anatol N.Kirillov, Toshiaki Maeno
    • Journal Title

      Publ.RIMS

      Volume: 48 Issue: 1 Pages: 215-228

    • DOI

      10.2977/prims/67

    • NAID

      110008906986

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A mathematical problem for security analysis of hash functions and pseudorandom generators2011

    • Author(s)
      K.Nuida, T.Abe, S.Kaji, T.Maeno, Y.Numata
    • Journal Title

      Advances in Information and Computer Security, LNCS

      Volume: 7,038 Pages: 144-160

    • DOI

      10.1007/978-3-642-25141-2_10

    • NAID

      120007100293

    • ISBN
      9783642251405, 9783642251412
    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Strong Lefschetz elements of the coinvariant rings of finite Coxeter groups2011

    • Author(s)
      T.Maeno, Y.Numata, A.Wachi
    • Journal Title

      Algebras and Representation Theory

      Volume: ((掲載確定) 電子媒体)

    • NAID

      130008004821

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Presentation] Finite geometry and the Lefschetz property2013

    • Author(s)
      Toshiaki Maeno
    • Organizer
      第35回可換環論シンポジウム
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2013 Annual Research Report
  • [Presentation] The Fomin-Kirillov quadratic algebra and its affinization2013

    • Author(s)
      Toshiaki Maeno
    • Organizer
      Bethe Ansatz, Quantum Groups and Beyond
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Alcove pathとaffine Weyl群上の非可換微分構造2011

    • Author(s)
      前野俊昭
    • Organizer
      日本数学会年会・無限可積分系セッション・特別講演
    • Place of Presentation
      早稲田大学
    • Year and Date
      2011-03-23
    • Related Report
      2010 Annual Research Report
  • [Book] The Lefschetz properties2013

    • Author(s)
      T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi, J. Watanabe
    • Publisher
      Springer
    • Related Report
      2013 Annual Research Report

URL: 

Published: 2010-08-23   Modified: 2023-03-16  

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