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Schubert calculus in quantum K theory

Research Project

Project/Area Number 18K03261
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionWaseda University (2020-2021)
Okayama University of Science (2018-2019)

Principal Investigator

Ikeda Takeshi  早稲田大学, 理工学術院, 教授 (40309539)

Co-Investigator(Kenkyū-buntansha) 松村 朝雄  岡山理科大学, 理学部, 講師 (80755223)
Project Period (FY) 2018-04-01 – 2022-03-31
Project Status Completed (Fiscal Year 2021)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2021: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords量子 K 環 / アフィン・グラスマン多様体 / Peterson 同型 / ヒルベルト・サミュエル重複度 / ヴェクシラリ / シューベルト多様体 / 量子 K 理論 / グラスマン多様体 / 量子K理論 / シューベルト・カルキュラス / Grothendieck 多項式 / K-Peterson 同型 / シューベルト類 / ピーターソン同型 / アフィングラスマン多様体 / 可積分系
Outline of Final Research Achievements

We study Schubert calculus of quantum K-theory ring of generalized flag varieties. In type A, we succeeded in describing the correspondence of Schubert classes given by K-theoretic Peterson isomorphism, as the correspondence between concrete polynomial representatives of the Schubert classes.

In a flag variety of classical type, we study singularity of points on a Schubert variety associated with a vexillary element. We proved a combinatorial formula for the multiplicities.

Academic Significance and Societal Importance of the Research Achievements

量子 K 理論におけるシューベルト・カルキュラスは,19世期に展開された「数え上げ幾何学」の自然な発展であり,構造定数の正値性など,良い性質を持っている.そのような性質を保ったまま,さらに一般的な「コホモロジー理論」でシューベルト・カルキュラスを展開することはできないと考えられるいくつかの理由がある.その意味で,シューベルト・カルキュラスにおける最終的な目標である.今回,このような問題へのアプローチとして K 理論的 Peterson 同型を非常に明示的に与えたことは今後の発展への突破口になる.

Report

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

    (19 results)

All 2022 2020 2019 2018 Other

All Int'l Joint Research (4 results) Journal Article (3 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 2 results,  Open Access: 1 results) Presentation (10 results) (of which Int'l Joint Research: 8 results,  Invited: 9 results) Book (1 results) Remarks (1 results)

  • [Int'l Joint Research] Ohio state university(米国)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] Da Lat university(ベトナム)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] Ajou University(韓国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Virginia Tech University(米国)

    • Related Report
      2018 Research-status Report
  • [Journal Article] Double Grothendieck Polynomials for Symplectic and Odd Orthogonal Grassmannians2020

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

      Journal of Algebra

      Volume: 546 Pages: 294-314

    • Related Report
      2019 Research-status Report
  • [Journal Article] Multiplicities of Schubert varieties in the symplectic flag variety2019

    • Author(s)
      D. Andeson, T. Ikeda, M. Jeon, R. Kawago
    • Journal Title

      S'eminaire Lotharingien de Combinatoire

      Volume: 82B

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Multiplicities of Schubert varieties in the symplectic flag variety2019

    • Author(s)
      D. Anderson, T. Ikeda, M. Jeon, R. Kawago
    • Journal Title

      Proceedings of the 31st Conference on Formal Power Series and Algebraic Combinatorics (Ljubljana) 2019

      Volume: -- Pages: 1-12

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] K 理論的シューベルト・カルキュラス2022

    • Author(s)
      池田岳
    • Organizer
      日本数学会2022年度年会 関数解析分科会 総合講演
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] Quantum K-theory of the Grassmannians, duality and its applications2019

    • Author(s)
      T. Ikeda
    • Organizer
      Representation Theory of Algebraic Groups and Quantum Groups
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Set-valued decomposition tableaux2019

    • Author(s)
      T. Ikeda
    • Organizer
      Crystals and Their Generalizations(大阪市立大学)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Relativistic Toda lattice and $K$-theoretic Peterson isomorphism2019

    • Author(s)
      Relativistic Toda lattice and $K$-theoretic Peterson isomorphism
    • Organizer
      Rikkyo MathPhys 2019\/}, 立教大学
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Relativistic Toda lattice and $K$-theoretic Peterson isomorphism2018

    • Author(s)
      T. Ikeda
    • Organizer
      Quantum $K$-theory and related topics, Korea Institute for Advanced Study
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Schubert calculus of Isotropic Grassmannians2018

    • Author(s)
      T. Ikeda
    • Organizer
      Seminar, Sun Yat-sen university, Guangzhou, China
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] シューベルト多様体の特異点の重複度に関する予想2018

    • Author(s)
      池田岳.川合遼太郎
    • Organizer
      セミナー,岡山大学理学部
    • Related Report
      2018 Research-status Report
  • [Presentation] $K$-theoretic Peterson isomorphism and its applications2018

    • Author(s)
      T. Ikeda
    • Organizer
      Hessenberg 集会 2018 in Osaka(大阪市立大学)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] $K$-theory Schubert calculus of the maximal orthogonal Grassmannian and set-valued decomposition tableaux2018

    • Author(s)
      T. Ikeda
    • Organizer
      Geometry, Combinatorics and Integral Systems Seminar(Ohio State University)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] $K$-theory Schubert calculus of the maximal orthogonal Grassmannian and set-valued decomposition tableaux2018

    • Author(s)
      T. Ikeda
    • Organizer
      Algebra Seminar(Virginia Tech)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Book] 数え上げ幾何学講義2018

    • Author(s)
      池田 岳
    • Total Pages
      280
    • Publisher
      東京大学出版会
    • ISBN
      9784130613125
    • Related Report
      2018 Research-status Report
  • [Remarks] 池田岳

    • URL

      https://www.xmath.ous.ac.jp/~ike/

    • Related Report
      2018 Research-status Report

URL: 

Published: 2018-04-23   Modified: 2023-01-30  

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