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2007 Fiscal Year Final Research Report Summary

Remark on modular representations of non-commutative algebraic systems

Research Project

Project/Area Number 16540023
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKyoto University

Principal Investigator

ARIKI Susumu  Kyoto University, Research Institute for Mathematical Sciences, Associate Professor (40212641)

Project Period (FY) 2004 – 2007
KeywordsHecke algebra / Modular representation / Kashiwara crystal / path model / degenerate BMW algebra / modular branching rule / DJM conjecture
Research Abstract

There are several kinds of algebras that appear in the studies of algebraic groups, quantum groups and conformal field theory As we may carry out detailed analysis of these algebras through applying various kinds of methods, there exists a research field which we might call "Special Noncommutative Algebras". Based on our 'previous research on modular representations of Hecke algebras, we set our goals in this research project in two themes: the first is development of our techniques to new algebras, and the second is to solve open problems in modular representation theory of Hecke algebras. For the first goal, we picked up degenerate affine BMW algebras defined over arbitrary algebraically closed field, and succeeded in constructing all the irreducible finite dimensional representations. This is a joint work with Mathas and Rui. This work influenced several other succeeding research on affine BMW algebras by other researchers. During the period, Rouquier showed that Cherednik algebras … More provide quasihereditary covers of cyclotomic Hecke algebras defined over the field of complex numbers. Thus, we have a chance to categorify Fock spaces. This gives a broad perspective which generalizes the head investigator's decomposition number theorem. By the above mentioned research developments, we have shifted to research which is more closely tied with representation theory of conformal field theory in the last year of the project, and we have obtained new insights for next research project. For the second goal, the head investigator has proved the modular branching rule for cyclotomic Hecke algebras, which was mentioned in the research proposal as one of the expected achievements. We have also settled a conjecture by Dipper, James and Murphy which has been open for 12 years. This is a joint work with Jacon. Recall that the Mullineux conjecture in the representation theory of the symmetric group (and the Hecke algebras of type A) had been open for many years, and it was finally settled by Kleshchev (and Brundan) in 90's, which was a big achievement. By applying Littelmann's path model to representation theory of Hecke algebras, we have obtained completely new description of the famous Mullineux map. Namely, the Mullineux map is always given by transpose of partitions even for non semi-simple Hecke algebras, if we work in the path model. The head investigator has published 13 papers (of which 8 papers are refereed, 1 is translation of a refereed paper) and presented 15 talks on the above results and results on the representation type of Hecke algebras of classical type during the research period. Less

  • Research Products

    (56 results)

All 2008 2007 2006 2005 2004 Other

All Journal Article (26 results) (of which Peer Reviewed: 9 results) Presentation (30 results)

  • [Journal Article] On the tensor product of two basic representations of $U_v(\hat{sl}_e)$2008

    • Author(s)
      Susumu Ariki, Victor Kreiman, Shunsuke Tsuchioka
    • Journal Title

      Advances in Math. (accepted)(掲載確定)

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Modular representations of Hecke algebras of classical type2007

    • Author(s)
      Susumu Ariki
    • Journal Title

      Sugaku Expositions 20

      Pages: 15,41

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Modular representations of Hecke algebras of classical type2007

    • Author(s)
      Susumu, Ariki
    • Journal Title

      Sugaku Expositions 20

      Pages: 15-41

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Algebras arising from Schur-Weyl type dualities2006

    • Author(s)
      Susumu Ariki
    • Journal Title

      Proc. Ring Theory and Represent. Theory 38

      Pages: 1,10

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Cyclotomic Nazarov-Wenzl algebras2006

    • Author(s)
      Susumu Ariki, Andrew Mathas and Hebing Rui
    • Journal Title

      Nagoya Math. J. 182

      Pages: 47,134

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] 巡回Nazarov-Wenzl代数のセル基底の構成2006

    • Author(s)
      有木 進
    • Journal Title

      数理解析研究所講究録 1476

      Pages: 145,151

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Proof of the modular branching rule for cyclotomic Hecke algebras2006

    • Author(s)
      Susumu Ariki
    • Journal Title

      J. Algebra 306

      Pages: 290,300

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Algebras arising from Schur-Weyl type dualities (Japanese)2006

    • Author(s)
      Susumu, Ariki
    • Journal Title

      Proc., Ring Theory and Represent. Theory 38

      Pages: 1-10

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Andrew Mathas and Hebing Rui, Cyclotomic Nazarov-Wenzl algebras2006

    • Author(s)
      Susumu, Ariki
    • Journal Title

      Nagoya Math. J 182

      Pages: 47-134

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Construction of cellular bases of Nazarov-Wenzl algebras (Japanese)2006

    • Author(s)
      Susumu, Ariki
    • Journal Title

      RIMS kokyuroku 1476

      Pages: 145-151

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Proof of the modular branching rule for cyclo-tomic Hecke algebras2006

    • Author(s)
      Susumu, Ariki
    • Journal Title

      J. Algebra 306

      Pages: 290-300

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Hecke algebras of classical type and their representation type2005

    • Author(s)
      Susumu Ariki
    • Journal Title

      Proc. London Math. Soc. (3) 91

      Pages: 355,413

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Modular representation theory of Hecke algebras, a survey2005

    • Author(s)
      Susumu Ariki
    • Journal Title

      Contemp. Math. 392

      Pages: 1,14

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] 巡回ヘッケ環,古典型有限群からCherednik代数まで2005

    • Author(s)
      有木 進
    • Journal Title

      代数学シンポジウム報告集 50

      Pages: 184,194

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Hecke algebras of classical type and their represent tation type2005

    • Author(s)
      Susumu, Ariki
    • Journal Title

      Proc. London Math. Soc. (3) 91

      Pages: 355-413

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Modular representation theory of Hecke algebras, a survey2005

    • Author(s)
      Susumu, Ariki
    • Journal Title

      Contemp. Math 392

      Pages: 1-14

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] From cyclotomic Hecke algebras, finite groups of classical type to Cherednik algebras (Japanese)2005

    • Author(s)
      Susumu, Ariki
    • Journal Title

      Proc. Algebra Symp 50

      Pages: 184-194

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Hecke algebras with a finite number of indecomposable modules2004

    • Author(s)
      Susumu Ariki and Andrew Mathas
    • Journal Title

      Adv. Stud. Pure Math. 40

      Pages: 17,25

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] 古典型Hecke環のモジュラー表現2004

    • Author(s)
      有木 進
    • Journal Title

      数学 56

      Pages: 113,136

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] 古典型Hecke環の表現型の決定2004

    • Author(s)
      有木 進
    • Journal Title

      数理解析研究所講究録 1382

      Pages: 74,79

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Hecke algebras with a finite number of indecomposable modules2004

    • Author(s)
      Susumu, Ariki, Andrew, Mathas
    • Journal Title

      Adv. Stud. Pure Math 40

      Pages: 17-25

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Modular representations of Hecke algebras of classical type (Japanese)2004

    • Author(s)
      Susumu, Ariki
    • Journal Title

      Sugaku 56

      Pages: 113-136

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Determination of the representation type of Hecke algebras (Japanese)2004

    • Author(s)
      Susumu, Ariki
    • Journal Title

      RIMS kokyuroku 1382

      Pages: 74-79

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Dipper-James-Murphy's conjecture for Hecke algebras of type B_n

    • Author(s)
      Susumu Ariki, Nicolas Jacon
    • Journal Title

      Progress in Math. (accepted)(印刷中)(掲載確定)

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Victor Kreiman and Shunsuke Tsuchioka, On the tensor product of two basic representations of $U v(\hat{sl}_e)$

    • Author(s)
      Susumu, Ariki
    • Journal Title

      Advances in Math. accepted

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Dipper-James-Murphy's conjecture for Hecke algebras of type B_n

    • Author(s)
      Susumu, Ariki, Nicolas, Jacon
    • Journal Title

      Progress in Math. accepted

    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Two applications of Littelmann's path model2008

    • Author(s)
      Susumu Ariki
    • Organizer
      Lie Theory
    • Place of Presentation
      バークレー,アメリカ
    • Year and Date
      2008-03-14
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Two applications of Littelmann's path model2008

    • Author(s)
      Susumu, Ariki
    • Organizer
      Lie Theory
    • Place of Presentation
      Berkeley USA
    • Year and Date
      2008-03-14
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Modular representation theory of Hecke algebras through categorification and its combinatorialization(I),(II),(III)2007

    • Author(s)
      Susumu Ariki
    • Organizer
      Categorification, Quantization and Clusters
    • Place of Presentation
      北京,中国
    • Year and Date
      20070910-13
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Modular representation theory of Hecke algebras through categorification and its combinatorialization (I),(II),(III)2007

    • Author(s)
      Susumu, Ariki
    • Organizer
      Categorification, Quantization and Clusters
    • Place of Presentation
      Beijing, China
    • Year and Date
      20070910-13
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Finite dimensional Hecke algebras(I),(II),(III)2007

    • Author(s)
      Susumu Ariki
    • Organizer
      ICRA XII(The 12th Int. Conf. on Repr. of Alg.)
    • Place of Presentation
      トルン,ポーランド
    • Year and Date
      20070816-18
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Finite dimensional Hecke algebras (I),(II),(III)2007

    • Author(s)
      Susumu, Ariki
    • Organizer
      ICRA XII (The 12th Int. Conf. on Repr. of Alg.)
    • Place of Presentation
      Torun, Poland
    • Year and Date
      20070816-18
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Crystal theory and Hecke algebra of type B_n2007

    • Author(s)
      Susumu Ariki
    • Organizer
      ICRT IV(The 4th Int. Conf. on Repr. Theory)
    • Place of Presentation
      ラサ,中国
    • Year and Date
      2007-07-17
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Crystal theory and Hecke algebra of type B_n2007

    • Author(s)
      Susumu, Ariki
    • Organizer
      ICRT IV (The 4th Int. Conf. on Repr. Theory)
    • Place of Presentation
      Lhasa, China
    • Year and Date
      2007-07-17
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] On a conjecture by Dipper, James and Murphy2007

    • Author(s)
      Susumu Ariki
    • Organizer
      GL07: Geometry and Lie Theory
    • Place of Presentation
      シドニー,オーストラリア
    • Year and Date
      2007-07-10
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] On a conjecture by Dipper, James and Murphy2007

    • Author(s)
      Susumu, Ariki
    • Organizer
      GL07 : Geometry and Lie Theory
    • Place of Presentation
      Sydney, Australia
    • Year and Date
      2007-07-10
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Algebras arising from Schur-Weyl type dualities, (I)(II)(III)2006

    • Author(s)
      Susumu, Ariki
    • Organizer
      The 38th Symposium on Ring Theory and Representation Theory
    • Place of Presentation
      Aichi, Japan
    • Year and Date
      20060902-04
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Tensor product of two basic representations of $U_v(\hat{sl}_e)$2006

    • Author(s)
      Susumu Ariki
    • Organizer
      Affine Hecke Algebras, The Langlands Program, Conformal Field Theory and Matrix Models
    • Place of Presentation
      CIRM, Luminy,フランス
    • Year and Date
      2006-06-19
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Tensor product of two basic representations of U v(\hat{sl}_e), Affine Hecke Algebras2006

    • Author(s)
      Susumu, Ariki
    • Organizer
      The Langlands Program, Conformal Field Theory and Matrix Models
    • Place of Presentation
      Luminy, France
    • Year and Date
      2006-06-19
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Non-recursive characterization of Kleshchev bipartitions2006

    • Author(s)
      Susumu Ariki
    • Organizer
      Representation Theory of Algebraic Groups and Quantum Groups 06
    • Place of Presentation
      名古屋大学
    • Year and Date
      2006-06-15
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Non-recursive characterization of Kleshchev bipartitions2006

    • Author(s)
      Susumu, Ariki
    • Organizer
      Representation Theory of Algebraic Groups and Quantum Groups 06
    • Place of Presentation
      Nagoya, Japan
    • Year and Date
      2006-06-15
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Algebras arising from Schur-Weyl type dualities(I)(II)(III)2005

    • Author(s)
      有木 進
    • Organizer
      第38回環論および表現論シンポジウム
    • Place of Presentation
      愛知工業大学
    • Year and Date
      20050902-03-04
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Irreducible representations of affine Wenzl algebras2005

    • Author(s)
      Susumu Ariki
    • Organizer
      TMS-AMS joint meeting
    • Place of Presentation
      東海大学(台中市),台湾
    • Year and Date
      2005-12-15
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Irreducible representations of affine Wenzl algebras2005

    • Author(s)
      Susumu, Ariki
    • Organizer
      TMS-AMS joint meeting
    • Place of Presentation
      Taichung, Taiwan
    • Year and Date
      2005-12-15
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] 巡回Nazarov-Wenzl代数のセル基底の構成2005

    • Author(s)
      有木 進
    • Organizer
      代数的組み合わせ論とその周辺
    • Place of Presentation
      数理解析研究所
    • Year and Date
      2005-10-05
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Construction of cellular basis of cyclotomic Nazarov-Wenzl algebras2005

    • Author(s)
      Susumu, Ariki
    • Organizer
      Algebraic Combinatorics and Related Topics
    • Place of Presentation
      RIMS,Japan
    • Year and Date
      2005-10-05
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] 巡回Hecke環,古典型有限群からCherednik代数まで2005

    • Author(s)
      有木 進
    • Organizer
      第50回代数学シンポジウム
    • Place of Presentation
      徳島大学
    • Year and Date
      2005-08-05
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] From cyclotomic Hecke algebras, finite groups of classical type to Cherednik algebras2005

    • Author(s)
      Susumu, Ariki
    • Organizer
      The 50th Algebra Symposium
    • Place of Presentation
      Tokushima, Japan
    • Year and Date
      2005-08-05
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Cyclotomic Wenzl Algebras2005

    • Author(s)
      Susumu Ariki
    • Organizer
      Algebraic Groups and Finite Reductive Groups
    • Place of Presentation
      ローザンヌ,スイス
    • Year and Date
      2005-06-13
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Cyclotomic Wenzl Algebras2005

    • Author(s)
      Susumu, Ariki
    • Organizer
      Algebraic Groups and Finite Reduc-tive Groups
    • Place of Presentation
      Lausanne Switzerland
    • Year and Date
      2005-06-13
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Recent advances in modular representation theory of Hecke algebras (I) (II) (III)2004

    • Author(s)
      Susumu, Ariki
    • Organizer
      Infinite-Dimensional Aspects of Repre-sentation Theory and Applications
    • Place of Presentation
      Charlottesville, USA
    • Year and Date
      20040518-21
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Recent advances in modular representation theory of Hecke algebras(I)(II)(III)2004

    • Author(s)
      Susumu Ariki
    • Organizer
      Infinite-Dimensional Aspects of Representation Theory and Applications
    • Place of Presentation
      シャーロッテビル,アメリカ
    • Year and Date
      20040518-19-21
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Representation type of Hecke algebras2004

    • Author(s)
      Susumu Ariki
    • Organizer
      Nilpotent Orbits and Representation Theory
    • Place of Presentation
      河口湖富士桜荘
    • Year and Date
      2004-09-09
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Representation type of Hecke algebras2004

    • Author(s)
      Susumu, Ariki
    • Organizer
      NORTH 6(Nilpotent Orbits and Rep-resentation Theory)
    • Place of Presentation
      Kawaguchiko, Japan
    • Year and Date
      2004-09-09
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Representation type of Hecke algebras and the Poincar\'e polynomial2004

    • Author(s)
      Susumu Ariki
    • Organizer
      Representations of Algebraic Groups, Quantum Groups, and Lie Algebras
    • Place of Presentation
      スノーバード,アメリカ
    • Year and Date
      2004-07-15
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Representation type of Hecke algebras and the Poincare polynomial2004

    • Author(s)
      Susumu, Ariki
    • Organizer
      Representations of Algebraic Groups, Quantum Groups, and Lie Algebras
    • Place of Presentation
      Snowbird, USA
    • Year and Date
      2004-07-15
    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2010-02-04  

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