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Schubert geometry and special polynomials

Research Project

Project/Area Number 20540053
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionOkayama University of Science

Principal Investigator

IKEDA Takeshi  Okayama University of Science, 理学部, 准教授 (40309539)

Co-Investigator(Kenkyū-buntansha) NARUSE Hiroshi  岡山大学, 教育学部, 教授 (20172596)
Co-Investigator(Renkei-kenkyūsha) OHMOTO Toru  北海道大学, 大学院・理学研究院, 准教授 (20264400)
NAKAGAWA Masaki  香川高等専門学校, 講師 (50370036)
Project Period (FY) 2008 – 2010
Project Status Completed (Fiscal Year 2010)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2010: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2009: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2008: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywordsシューベルト類 / 同変コホモロジー / 特殊多項式 / 旗多様体 / グラスマン多様体 / ピエリ型公式 / 超対称多項式 / 同変K理論 / シューベルト幾何 / 集合値shifted tableau / Schubert幾何 / コホモロジー / Schur関数
Research Abstract

For the flag variety of classical Lie groups, we introduced special polynomials representing the Schubert classes in torus equivariant cohomology ring. We established some fundamental properties for these polynomials. In order to extend this result to torus equivariant K-theory, we introduced special family of polynomials (K-theoretic Q-and P-functions) representing the Schubert classes in the torus equivariant K-theory of classical Grassmannian varieties. We also developed combinatorics of these polynomials. In particular, we discussed a relationship to excited Young diagrams and shifted set-valued tableaux. We introduced a Robinson-Schensted type algorithm for these tableaux. As an application of this algorithm, we proved Pieri type formulas for the K-theoretic Q-functions.

Report

(4 results)
  • 2010 Annual Research Report   Final Research Report ( PDF )
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • Research Products

    (52 results)

All 2011 2010 2009 2008 2007 Other

All Journal Article (12 results) (of which Peer Reviewed: 12 results) Presentation (40 results)

  • [Journal Article] Double Schubert polynomials for the classical groups2011

    • Author(s)
      Takeshi Ikeda, Leonardo, Mihalcea, Hiroshi Naruse
    • Journal Title

      Adv.Math 226

      Pages: 840-886

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Bumping algorithm for set-valued shifted tableaux (extended abstract for FPSAC 2011, Reykjavik)2011

    • Author(s)
      Takeshi Ikeda, Yasuhide Numata, Hiroshi Naruse
    • Journal Title

      Discrete Mathematics and Theoretical Computer Science Proceeding volume for fpsac(online 掲載確定)

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Double Schubert polynomials for the classical groups2011

    • Author(s)
      Takeshi Ikeda, Leonardo Mihalcea, Hiroshi Naruse
    • Journal Title

      Advances in Mathematics

      Volume: 226 Pages: 840-886

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Double Schubert polynomials of classical type and Excited Young diagrams RIMS Kokyuroku Bessatsu2009

    • Author(s)
      Takeshi Ikeda, Hiroshi Naruse
    • Journal Title

      B11

      Pages: 87-100

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Excited Young diagrams and equivariant Schubert calculus Trans.2009

    • Author(s)
      Takeshi Ikeda, Hiroshi Naruse
    • Journal Title

      Amer.Math.Soc 361

      Pages: 5193-5221

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Double Schubert polynomials of classical type and Excited Young diagrams2009

    • Author(s)
      Takeshi Ikeda, Hiroshi Naruse
    • Journal Title

      RIMS Bessatsu B11

      Pages: 87-100

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Excited Young diagrams and equivariant Schubert calculus2009

    • Author(s)
      Takeshi Ikeda, Hiroshi Naruse
    • Journal Title

      Trans.Amer.Math.Soc.361 361

      Pages: 5193-5221

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Excite Young diagrams and Eguivariant schubat Cakcykys2009

    • Author(s)
      T. Ikeda, H. Naruse
    • Journal Title

      Trans. Amer. Mafh. Soc.

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Double Schueat polyaowiaes of dadicae tyesand Excifed young diagraws2009

    • Author(s)
      T. Iheda, H. Naruse
    • Journal Title

      RIMS Kokyuroku Bessatsu B11

      Pages: 87-100

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Mixed expansion formula for the rectangular Schur functions and the affine Lie algebra2008

    • Author(s)
      H.Mizhkawa, T.Nakajima, T.Ikeda, H.-F.Yamada
    • Journal Title

      Adv.Appl.Math 40

      Pages: 514-535

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Mixed Expansion forwuda for the necfangulan Sclw functius and the oBtine Lie aloebra2008

    • Author(s)
      H. Mizukawa, T. Nakajima, T. Ikeda, H. -F. Yamada
    • Journal Title

      Adv. Appe. Mafh 40

      Pages: 514-535

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Bumping algorithm for set-valued shifted tableaux (extended abstract for FPSAC 2011, Reykjavik)

    • Author(s)
      Takeshi Ikeda, Yasuhide Numata, Hiroshi Naruse
    • Journal Title

      Discrete Mathematics and Theoretical Computer Science

      Volume: Proceeding volume for fpsac 2011(online 掲載確定)

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Presentation] K-nil-Hecke algebraについて2011

    • Author(s)
      池田岳
    • Organizer
      GKMグラフに関する勉強会
    • Place of Presentation
      大阪市立大学理学部
    • Year and Date
      2011-03-16
    • Related Report
      2010 Annual Research Report 2010 Final Research Report
  • [Presentation] K-theoretic Schubert calculus for Lagrangian Grassmannian港の西北2010

    • Author(s)
      池田岳
    • Organizer
      早稲田大学代数幾何学シンポジウム
    • Place of Presentation
      早稲田大学理工学部
    • Year and Date
      2010-11-11
    • Related Report
      2010 Final Research Report
  • [Presentation] K-theoreic Schubert calculus for Lagrangian Grassmannian2010

    • Author(s)
      池田岳
    • Organizer
      港の西北,早稲田大学代数幾何学シンポジウム
    • Place of Presentation
      早稲田大学理工学部
    • Year and Date
      2010-11-11
    • Related Report
      2010 Annual Research Report
  • [Presentation] Quantum Schubert vs affine Schubert via nilpotent Toda lattice2010

    • Author(s)
      池田岳
    • Organizer
      研究集会「表現論と組み合わせ論」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2010-10-19
    • Related Report
      2010 Annual Research Report 2010 Final Research Report
  • [Presentation] Boson-Fermion correspondence and KP-hierarchy I2010

    • Author(s)
      池田岳
    • Organizer
      II p-進佐藤理論と数論幾何
    • Place of Presentation
      東北大学大学院理学研究科
    • Year and Date
      2010-09-29
    • Related Report
      2010 Final Research Report
  • [Presentation] Boson-Fermion correspondence and KP-hierarchy I, II2010

    • Author(s)
      池田岳
    • Organizer
      p-進佐藤理論と数論幾何
    • Place of Presentation
      東北大学大学院理学研究科
    • Year and Date
      2010-09-29
    • Related Report
      2010 Annual Research Report
  • [Presentation] Affine GrassmannianのSchubert2010

    • Author(s)
      池田岳
    • Organizer
      多項式第2回「シューベルト・カルキュラスとその周辺」
    • Place of Presentation
      倉敷シーサイドホテル
    • Year and Date
      2010-09-22
    • Related Report
      2010 Final Research Report
  • [Presentation] Affine Grassmannian の Schubert 多項式2010

    • Author(s)
      池田岳
    • Organizer
      第2回「シューベルト・カルキュラスとその周辺」
    • Place of Presentation
      倉敷シーサイドホテル
    • Year and Date
      2010-09-22
    • Related Report
      2009 Annual Research Report
  • [Presentation] Review on Koszul duality2010

    • Author(s)
      池田岳
    • Organizer
      第13回代数群と量子群の表現論研究集会
    • Place of Presentation
      すいとぴあ江南
    • Year and Date
      2010-06-06
    • Related Report
      2010 Annual Research Report 2010 Final Research Report
  • [Presentation] Schur Q-関数の K-理論的類似とそのPieri 型公式2010

    • Author(s)
      池田岳,成瀬弘,沼田泰英
    • Organizer
      日本数学会2010年度年会
    • Place of Presentation
      慶応大学矢上キャンパス
    • Year and Date
      2010-03-27
    • Related Report
      2010 Final Research Report
  • [Presentation] Schur Q-関数のK-理論的類似とその Pieri 型公式2010

    • Author(s)
      池田岳・成瀬弘・沼田泰英
    • Organizer
      日本数学会2010年度年会
    • Place of Presentation
      慶応大学矢上キャンパス
    • Year and Date
      2010-03-27
    • Related Report
      2009 Annual Research Report
  • [Presentation] 同変シューベルト幾何における最近のトピックス2010

    • Author(s)
      池田岳
    • Organizer
      第53回Encounter with Mathematics
    • Place of Presentation
      中央大学理工学部
    • Year and Date
      2010-03-09
    • Related Report
      2010 Final Research Report
  • [Presentation] 同変シューベルト幾何における最近のトピックス2010

    • Author(s)
      池田岳
    • Organizer
      第53回 Encounter with Mathematics
    • Place of Presentation
      中央大学理工学部
    • Year and Date
      2010-03-09
    • Related Report
      2009 Annual Research Report
  • [Presentation] シューベルト幾何の八方出汁2010

    • Author(s)
      池田岳
    • Organizer
      第53回Encounter with Mathematics
    • Place of Presentation
      中央大学理工学部
    • Year and Date
      2010-03-08
    • Related Report
      2010 Final Research Report
  • [Presentation] シューベルト幾何の八方出汁2010

    • Author(s)
      池田岳
    • Organizer
      第53回 Encounter with Mathematics
    • Place of Presentation
      中央大学理工学部
    • Year and Date
      2010-03-08
    • Related Report
      2009 Annual Research Report
  • [Presentation] 冪零戸田格子とアフィングラスマン多様体2010

    • Author(s)
      池田岳
    • Organizer
      Workshop on Representation Theory
    • Place of Presentation
      東京大学玉原国際セミナーハウス
    • Related Report
      2010 Annual Research Report 2010 Final Research Report
  • [Presentation] BGGによるKoszul双対性の解説2010

    • Author(s)
      池田岳
    • Organizer
      Seminar on Koszul duality
    • Place of Presentation
      岡山大学大学院自然科学研究棟
    • Related Report
      2010 Annual Research Report 2010 Final Research Report
  • [Presentation] 古典型旗多様体の同変K理論におけるシューベルト類に付随する多項式について2009

    • Author(s)
      成瀬弘
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-27
    • Related Report
      2010 Final Research Report 2009 Annual Research Report
  • [Presentation] 表現論と組合せ論2009

    • Author(s)
      池田岳
    • Organizer
      Schur Q関数のK-理論的拡張と等法的グラスマン多様体数理研研究集会
    • Place of Presentation
      北海道大学
    • Year and Date
      2009-08-25
    • Related Report
      2010 Final Research Report
  • [Presentation] Schur Q関数のK-理論的拡張と等法的グラスマン多様体2009

    • Author(s)
      池田岳
    • Organizer
      数理研研究集会「表現論と組合せ論」
    • Place of Presentation
      北海道大学
    • Year and Date
      2009-08-25
    • Related Report
      2009 Annual Research Report
  • [Presentation] シューベルト多項式2009

    • Author(s)
      池田岳
    • Organizer
      の現在第56回トポロジーシンポジウム
    • Place of Presentation
      北海道大学
    • Year and Date
      2009-08-10
    • Related Report
      2010 Final Research Report
  • [Presentation] 「シューベルト多項式」の現在2009

    • Author(s)
      池田岳
    • Organizer
      第56回 トポロジーシンポジウム
    • Place of Presentation
      北海道大学
    • Year and Date
      2009-08-10
    • Related Report
      2009 Annual Research Report
  • [Presentation] SchurのQ多項式のK-理論的拡張について2009

    • Author(s)
      池田岳
    • Organizer
      東北大学数学教室談話会
    • Place of Presentation
      東北大学
    • Year and Date
      2009-06-08
    • Related Report
      2010 Final Research Report
  • [Presentation] Schur のQ多項式のK-理論的拡張について2009

    • Author(s)
      池田岳
    • Organizer
      東北大学 数学教室 談話会
    • Place of Presentation
      東北大学
    • Year and Date
      2009-06-08
    • Related Report
      2009 Annual Research Report
  • [Presentation] Schubert geometry of the flag varieties of classical types2008

    • Author(s)
      池田岳
    • Organizer
      Conference on Combinatorics and Representation theory
    • Place of Presentation
      名古屋大学
    • Year and Date
      2008-09-05
    • Related Report
      2010 Final Research Report
  • [Presentation] Schubert Geometry of teh flag varieties fo classical types2008

    • Author(s)
      池田岳
    • Organizer
      Conference on Covnbinaforics and pcpresentation Theory
    • Place of Presentation
      名古屋大学
    • Year and Date
      2008-09-05
    • Related Report
      2008 Annual Research Report
  • [Presentation] 古典型二重シューベルト多項式とその拡張について2008

    • Author(s)
      池田岳,成瀬弘
    • Organizer
      代数学シンポジウム
    • Place of Presentation
      盛岡市アイーナ
    • Year and Date
      2008-08-05
    • Related Report
      2010 Final Research Report
  • [Presentation] 古曲型二重シューペルト多項式とその拡張について2008

    • Author(s)
      池田岳成瀬弘
    • Organizer
      代数学シンポジウ厶
    • Place of Presentation
      盛岡市アイーナ
    • Year and Date
      2008-08-05
    • Related Report
      2008 Annual Research Report
  • [Presentation] Schubert 幾何とSchur多項式2008

    • Author(s)
      池田岳
    • Organizer
      可積分系ひ幾何学的アプローチ
    • Place of Presentation
      慶応大学
    • Year and Date
      2008-07-31
    • Related Report
      2008 Annual Research Report
  • [Presentation] 特殊関数としてのシューベルト多項式2008

    • Author(s)
      池田岳
    • Place of Presentation
      日九州可積分系セミナー
    • Year and Date
      2008-07-10
    • Related Report
      2010 Final Research Report
  • [Presentation] 特殊関教としてのシューベルト多項式2008

    • Author(s)
      池田岳
    • Organizer
      九州可積分系セミナー
    • Place of Presentation
      九州大学
    • Year and Date
      2008-07-10
    • Related Report
      2008 Annual Research Report
  • [Presentation] 特殊関数としてのシューベルト多項式2008

    • Author(s)
      池田岳
    • Organizer
      神戸可積分系セミナー
    • Year and Date
      2008-06-05
    • Related Report
      2010 Final Research Report
  • [Presentation] 特殊関教としてのシューベルト多項式2008

    • Author(s)
      池田岳
    • Organizer
      神戸可積分系セミナー
    • Place of Presentation
      神戸大学
    • Year and Date
      2008-06-05
    • Related Report
      2008 Annual Research Report
  • [Presentation] 旗多様体のトーラス同変シューベルト類2008

    • Author(s)
      池田岳
    • Organizer
      東京幾何セミナー
    • Place of Presentation
      東京工業大学
    • Year and Date
      2008-05-28
    • Related Report
      2010 Final Research Report
  • [Presentation] 旗多様体のトーラス同変シユベルト類2008

    • Author(s)
      池田岳
    • Organizer
      東京幾何セミナー
    • Place of Presentation
      東京工業大学
    • Year and Date
      2008-05-28
    • Related Report
      2008 Annual Research Report
  • [Presentation] 古典群の旗多様体と二重シューベルト多項式2008

    • Author(s)
      池田岳
    • Place of Presentation
      東北大学
    • Year and Date
      2008-04-15
    • Related Report
      2010 Final Research Report
  • [Presentation] 古典群の旗多様体とこ重シュベルト多項式2008

    • Author(s)
      池田岳
    • Organizer
      東北大学幾何セミナー
    • Place of Presentation
      東北大学
    • Year and Date
      2008-04-15
    • Related Report
      2008 Annual Research Report
  • [Presentation] 現代のシューベルト・カルキュラス2008

    • Author(s)
      池田岳
    • Organizer
      東北大学談話会
    • Place of Presentation
      東北大学
    • Year and Date
      2008-04-14
    • Related Report
      2010 Final Research Report
  • [Presentation] 現代のシューベルトカリキュラス2008

    • Author(s)
      池田岳
    • Organizer
      東北大学談話会
    • Place of Presentation
      東北大学
    • Year and Date
      2008-04-14
    • Related Report
      2008 Annual Research Report
  • [Presentation] Schubert幾何とSchur多項式2007

    • Author(s)
      池田岳
    • Organizer
      研究集会「可積分系への幾何学的アプローチ」
    • Place of Presentation
      慶応大学
    • Year and Date
      2007-07-31
    • Related Report
      2010 Final Research Report

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Published: 2008-04-01   Modified: 2016-04-21  

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