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Invariants of the Weyl groups of exceptional Lie groups and their applications

Research Project

Project/Area Number 18540106
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTakamatsu National College of Technology

Principal Investigator

NAKAGAWA Masaki  Takamatsu National College of Technology, 一般教育科, 講師 (50370036)

Co-Investigator(Kenkyū-buntansha) 三村 護  岡山大学, その他, 名誉教授 (70026772)
Project Period (FY) 2006 – 2008
Project Status Completed (Fiscal Year 2008)
Budget Amount *help
¥3,540,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥540,000)
Fiscal Year 2008: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2007: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2006: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywords不変式環 / Weyl 群 / 分類空間 / コホモロジー / 旗多様体 / Schubert 算法 / 等質空間 / Schubert算法 / Chow環 / 代数的組合せ論 / Schubert calculus / Wey1群 / 例外型Lie群 / Shubert多様体 / Weyl群
Research Abstract

コンパクト単連結単純Lie 群G をその極大トーラスT で割って得られる完全旗多様体G/T のコホモロジー環は, G のWeyl 群の極大トーラスT への作用に関する不変式環を用いて記述されるが(Borel 表示), これをもとにG が例外型Lie 群E_8 の場合に旗多様体E_8/T の整数係数コホモロジー環を決定した. さらに, 差分商作用素を計算するMaple プログラムを作成し, これを用いて, 全てのコンパクト単連結単純Lie 群G に対して, 対応する旗多様体G/T の整数係数コホモロジー環のSchubert 類による表示(Schubert 表示)を与えた. さらに, これらの結果とGrothendieck のremark をもとに, G が例外型Lie 群E_6, E_7, E_8 の場合に, 対応する複素Lie 群のChow 環を決定した.

Report

(4 results)
  • 2008 Annual Research Report   Final Research Report ( PDF )
  • 2007 Annual Research Report
  • 2006 Annual Research Report
  • Research Products

    (31 results)

All 2008 2007 Other

All Journal Article (14 results) (of which Peer Reviewed: 9 results) Presentation (17 results)

  • [Journal Article] A description based on Schubert classes of cohomology of flag manifolds2008

    • Author(s)
      M. Nakagawa
    • Journal Title

      Fundamenta Mathematicae 199

      Pages: 273-293

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] The Chow rings of the algebraic groups E_6, E_7, and E_82008

    • Author(s)
      M. Nakagawa
    • Journal Title

      京都大学数理解析研究所講究録 1581

      Pages: 92-102

    • Related Report
      2008 Final Research Report
  • [Journal Article] On the Rothenberg-Steenrod spectral sequence for the mod 2 cohomology of classifying space of spinor groups2008

    • Author(s)
      M. Kameko and M. Mimura
    • Journal Title

      Geometry and Topology Monographs 13

      Pages: 261-279

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Cartan models and cellular decompositions of symmetric Riemannian spaces2008

    • Author(s)
      H. Kadzisa and M. Mimura
    • Journal Title

      Topology and Its Applications 156

      Pages: 348-364

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] A description based on Schubert classes of cohomology of flag manifolds2008

    • Author(s)
      Masaki Nakagawa
    • Journal Title

      Fundamenta Mathematicae 199

      Pages: 273-293

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Chow rings of the algebraic groups E_6, E_7, and E_82008

    • Author(s)
      Masaki Nakagawa
    • Journal Title

      京都大学数理解析研究所講究録 1581

      Pages: 92-102

    • Related Report
      2007 Annual Research Report
  • [Journal Article] On the Lusternik-Schnirelmann category of symmetric spaces of classical type2008

    • Author(s)
      Mamoru Mimura and Kei Sugata
    • Journal Title

      Geometry & Topology Monographs 13

      Pages: 323-334

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the Rothenberg-Steenrod spectral sequence for the mod 2 cohomology of classifying spaces of spinor groups2008

    • Author(s)
      Masaki Kameko and Mamoru Mimura
    • Journal Title

      Geometry & Topology Monographs 13

      Pages: 261-279

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Mui invariants and Milnor operations2007

    • Author(s)
      M. Kameko and M. Mimura
    • Journal Title

      Geometry and Topology Monographs 11

      Pages: 107-140

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] On the Rothenberg-Steenrod spectral sequence for the mod 3 cohomology of the classifying space of the exceptional Lie group E_82007

    • Author(s)
      M. Kameko and M. Mimura
    • Journal Title

      Geometry and Topology Monographs 10

      Pages: 213-226

    • Related Report
      2008 Final Research Report
    • Peer Reviewed
  • [Journal Article] Morse-Bott functions and the Lusternik-Schnirelmann category2007

    • Author(s)
      Hiroyuki Kadzisa and Mamoru Mimura
    • Journal Title

      Proceedings of the 5th Sogan Geometry and Complex Analysis Symposium

      Pages: 25-56

    • Related Report
      2007 Annual Research Report
  • [Journal Article] Cartan models and cellular decompositions of symmetric spaces2007

    • Author(s)
      Hiroyuki Kadzisa and Mamoru Mimura
    • Journal Title

      Proceedings of East Asian Conference on Algebraic Topology

      Pages: 125-130

    • Related Report
      2007 Annual Research Report
  • [Journal Article] A description based on Schubert classes of cohomology of flag manifolds

    • Author(s)
      Masaki Nakagawa
    • Journal Title

      Fundamenta Mathematicaeに (掲載予定)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Mui invariants and Milonor operations

    • Author(s)
      Masaki Kameko, Mamoru Mimura
    • Journal Title

      Goemetry and Topology Monographs (印刷中)

    • Related Report
      2006 Annual Research Report
  • [Presentation] Schubert calculus in thecohomology rings of flag varieties2008

    • Author(s)
      M. Nakagawa
    • Organizer
      第55回トポロジーシンポジウム
    • Place of Presentation
      金沢市文化ホール
    • Year and Date
      2008-08-08
    • Related Report
      2008 Final Research Report
  • [Presentation] Schubert calculus in the cohomology rings of flag varieties2008

    • Author(s)
      中川征樹
    • Organizer
      第55回トポロジーシンポジウム
    • Place of Presentation
      金沢市文化ホール
    • Year and Date
      2008-08-08
    • Related Report
      2008 Annual Research Report
  • [Presentation] The Chow rings of complex algebraic groups2008

    • Author(s)
      M. Nakagawa
    • Organizer
      Conference on Algebraic and Geometric Topology
    • Place of Presentation
      Gdansk 大学
    • Year and Date
      2008-06-09
    • Related Report
      2008 Final Research Report
  • [Presentation] The Chow rings of complex algebraic groups2008

    • Author(s)
      Masaki Nakagawa
    • Organizer
      Conference on Algebraic and Geometric Topology
    • Place of Presentation
      Gdansk大学
    • Year and Date
      2008-06-09
    • Related Report
      2008 Annual Research Report
  • [Presentation] The integral cohomology rings of full flag manifolds2008

    • Author(s)
      M. Nakagawa
    • Organizer
      Workshop on Schubert calculus
    • Place of Presentation
      関西セミナーハウス
    • Year and Date
      2008-03-20
    • Related Report
      2008 Final Research Report 2007 Annual Research Report
  • [Presentation] Multiplicative structure of the Chow rings of projective homogeneous varieties2007

    • Author(s)
      M. Nakagawa
    • Organizer
      ホモトピー論シンポジウム
    • Place of Presentation
      金沢市観光会館
    • Year and Date
      2007-11-14
    • Related Report
      2008 Final Research Report 2007 Annual Research Report
  • [Presentation] 古典型対称空間の cone-decomposition2007

    • Author(s)
      M. Mimura
    • Organizer
      ホモトピー論シンポジウム
    • Place of Presentation
      金沢市観光会館
    • Year and Date
      2007-11-13
    • Related Report
      2008 Final Research Report
  • [Presentation] 古典型対称空間のcone-decomposition2007

    • Author(s)
      Mamoru Mimura
    • Organizer
      ホモトピー論シンポジウム
    • Place of Presentation
      金沢市観光会館
    • Year and Date
      2007-11-13
    • Related Report
      2007 Annual Research Report
  • [Presentation] Cone-decomposition of symmetric spaces of classical type2007

    • Author(s)
      M. Mimura
    • Organizer
      East Asian Conference on Algebraic Topology
    • Place of Presentation
      ソウル国立大学
    • Year and Date
      2007-11-03
    • Related Report
      2008 Final Research Report
  • [Presentation] Cone-decomposition of symmetric spaces of classical type2007

    • Author(s)
      Mamoru Mimura
    • Organizer
      East Asian Conference on Algebraic Topology
    • Place of Presentation
      Seoul National University (Korea)
    • Year and Date
      2007-11-03
    • Related Report
      2007 Annual Research Report
  • [Presentation] 代数群E_6, E_7, E_8 の Chow 環2007

    • Author(s)
      M. Nakagawa
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-22
    • Related Report
      2008 Final Research Report
  • [Presentation] 代数群E_6, E_7, E_8のChow環2007

    • Author(s)
      Masaki Nakagawa
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-22
    • Related Report
      2007 Annual Research Report
  • [Presentation] The Chow rings of the algebraic groups E_6 and E_7, and E_82007

    • Author(s)
      M. Nakagawa
    • Organizer
      有限群のコホモロジーとその周辺
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2007-08-30
    • Related Report
      2008 Final Research Report
  • [Presentation] The Chow rings of the algebraic groups E_6, E_7, and E_82007

    • Author(s)
      Masaki Nakagawa
    • Organizer
      有限群のコホモロジーとその周辺
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2007-08-30
    • Related Report
      2007 Annual Research Report
  • [Presentation] On the Lusternik-Schnirelmann category of Lie groups and homogeneous spaces2007

    • Author(s)
      Mamoru Mimura
    • Organizer
      Topology Seminar of KIAS
    • Place of Presentation
      Korean Institute for Advanced Study (Korea)
    • Year and Date
      2007-06-21
    • Related Report
      2007 Annual Research Report
  • [Presentation] Morse-Bott functions and the Lusternik-Schnirelmann category2007

    • Author(s)
      Mamoru Mimura
    • Organizer
      2007 Sogan Geometry and Complex Analysis Symposium
    • Place of Presentation
      Sogan University (Korea)
    • Year and Date
      2007-05-25
    • Related Report
      2007 Annual Research Report
  • [Presentation] On the Lusternik-Schnirelmann category of symmetric spaces of classical type2007

    • Author(s)
      Mamoru Mimura
    • Organizer
      韓国数学会年会
    • Place of Presentation
      Inha University (Korea)
    • Year and Date
      2007-04-21
    • Related Report
      2007 Annual Research Report

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Published: 2006-04-01   Modified: 2016-04-21  

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