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Construction of tropical R maps on geometric crystals and its applications to crystal bases

Research Project

Project/Area Number 22540031
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

NAKASHIMA Toshiki  上智大学, 理工学部, 教授 (60243193)

Co-Investigator(Kenkyū-buntansha) TSUZUKI Masao  上智大学, 理工学部, 准教授 (80296946)
GOMI Yasushi  上智大学, 理工学部, 准教授 (50276515)
筱田 健一 (篠田 健一)  上智大学, 理工学部, 教授 (20053712)
Project Period (FY) 2010-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
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: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords幾何結晶 / 結晶基底 / トロピカルR行列 / 多面体表示 / 一般化小行列式 / 単項式表示 / Bruhat cell / ワイル群 / 代数学 / 量子群 / トロピカルR写像 / 一般化行列式 / ブリューワーセル / クラスター代数 / 超離散化 / トロピカルR写像 / picture / 冪単結晶 / イプシロン系 / トロピカルRマップ
Outline of Final Research Achievements

We considered decorated geometric crystals, which are deeply related to tropical
R-maps on geometric crystals.For types A,B,C,D and G, we obtained several results on the relations with monomial realizations of crystal bases.In particular, decoration functions of decorated geometric crystals are expressed as a sum of monomials with positive coefficients via generalized minors, which means that there exist certain interesting relations between monomial realizations and polyhedral realizations of crystal bases and these results are all new.

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

    (28 results)

All 2014 2013 2012 2011 2010 Other

All Journal Article (14 results) (of which Peer Reviewed: 14 results,  Acknowledgement Compliant: 2 results) Presentation (14 results) (of which Invited: 7 results)

  • [Journal Article] Decorated Geometric Crystals and Polyhedral Realizations of type D_n2014

    • Author(s)
      Nakashima, Toshiki
    • Journal Title

      Contemporary Mathematics

      Volume: 623 Pages: 227-242

    • DOI

      10.1090/conm/623/12466

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Decorations on geometric crystals and monomial realizations of crystal bases for classical groups2014

    • Author(s)
      Nakashima, Toshiki
    • Journal Title

      Journal of Algebra

      Volume: 399 Pages: 712-769

    • DOI

      10.1016/j.jalgebra.2013.09.052

    • Related Report
      2014 Annual Research Report 2013 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] RSK-type correspondence of Pictures and Littlewood-Richardson Crystals2013

    • Author(s)
      Toshiki Nakashima and Miki Shimojo
    • Journal Title

      Tokyo Journal of Mathematics

      Volume: 36 Issue: 1 Pages: 113-130

    • DOI

      10.3836/tjm/1374497514

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Decorated Geometric Crystals, Polyhedral and Monomial Realizations of Crystal Bases2013

    • Author(s)
      Toshiki Nakashima
    • Journal Title

      Contemporary Mathematics

      Volume: 602 Pages: 143-163

    • DOI

      10.1007/978-1-4471-4863-0_12

    • ISBN
      9781447148623, 9781447148630
    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A_n^(1)-Geometric Crystal corresponding to Dynkin index i=2 and its ultra-discretization2013

    • Author(s)
      K. Misra and T.Nakashima
    • Journal Title

      Springer Proceedings in Mathematics & Statistics

      Volume: 40 Pages: 297-318

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Ultra-discretization of the D^(3)_4-Geometric Crystals to the G^(1)_2-Perfect Crystals2012

    • Author(s)
      M.Igarashi, Kailash C.Misra and T.Nakashima
    • Journal Title

      Pacific Journal of Mathematics

      Volume: 255 Pages: 117-152

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Gauss sums on finite groups2012

    • Author(s)
      Y. Gomi, T. Maeda and K. Shinoda
    • Journal Title

      Tokyo Journal of Mathematics

      Volume: 35 Pages: 165-179

    • Related Report
      2012 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Geometric Crystals on Flag varieties and Unipotent Subgroups of Classical Groups2011

    • Author(s)
      M.Igarashi, T.Nakashima
    • Journal Title

      Journal of Geometry and Physics

      Volume: 61 Issue: 11 Pages: 2267-2284

    • DOI

      10.1016/j.geomphys.2011.07.001

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Pictures and Littlewood-Richardson Crystals2011

    • Author(s)
      T.Nakashima, M.Shimojo
    • Journal Title

      Tokyo Jouranl of Mathematics

      Volume: 34 Pages: 493-506

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Admissible Pictures and Littlewood-Richardson Crystals2011

    • Author(s)
      T.Nakashima, M.Shimojo
    • Journal Title

      Communications in Algebra

      Volume: 39 Issue: 10 Pages: 3849-3865

    • DOI

      10.1080/00927872.2010.512590

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Tropical R maps and Affine Geometric Crystals2010

    • Author(s)
      M.Kashiwara, T.Nakashima, M.Okado
    • Journal Title

      Representation Theory

      Volume: 14 Pages: 446-509

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Epsilon Systems on Geometric Crystals of type A_n2010

    • Author(s)
      T.Nakashima
    • Journal Title

      Symmetry, Integrability and Geometry : Methods and Applications(SIGMA)

      Volume: 6 Pages: 023-14

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Ultra-discretization of the G^{(1)}_2-Geometric Crystals to the D^{(3)}_4-Perfect Crystals2010

    • Author(s)
      M.Igarashi, T.Nakashima
    • Journal Title

      Progress in Mathematics

      Volume: 284 Pages: 273-296

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Affine Geometric Crystal of type D^{(3)}_42010

    • Author(s)
      M.Igarashi, T.Nakashima
    • Journal Title

      Contemporary Mathematics

      Volume: 506 Pages: 215-226

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Presentation] Crystal Bases and Upper Cluster Algebras on Double Bruhat Cells2014

    • Author(s)
      Nakashima, Toshiki
    • Organizer
      Conference on Cluster Algebras and Representation Theory, CMC Thematic Program on Cluster Algebras in Mathematics and Physics
    • Place of Presentation
      Seoul, Korea
    • Year and Date
      2014-11-04 – 2014-11-07
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Upper Cluster Algebras on Double Bruhat Cells and Monomial Realizations of Crystal Bases2014

    • Author(s)
      Nakashima, Toshiki
    • Organizer
      ICM satellite Conference on Representation Theory and Related Topics
    • Place of Presentation
      Daegu, Korea
    • Year and Date
      2014-08-05 – 2014-08-10
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Cluster Variables on Double Bruhat cells and Monomial Realizations of Crystal Bases2014

    • Author(s)
      Nakashima, Toshiki
    • Organizer
      New Directions in Lie Theory, Combinatorial Representation Theory
    • Place of Presentation
      Montreal, Canada
    • Year and Date
      2014-04-19 – 2014-04-27
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Cluster Variables on Double Bruhat Cells and Monomial Realizations of Crystal Bases2014

    • Author(s)
      中島 俊樹
    • Organizer
      New Directions in Lie Theory
    • Place of Presentation
      Montreal, Canada
    • Related Report
      2013 Annual Research Report
    • Invited
  • [Presentation] Elementary Characters and Monomial Realizations of Crystal Base2011

    • Author(s)
      中島俊樹
    • Organizer
      Infinite Analysis 11 Winter School : Quantum cluster algebras and related topics, Graduate School of Science
    • Place of Presentation
      大阪大学(招待講演)
    • Year and Date
      2011-12-23
    • Related Report
      2011 Annual Research Report
  • [Presentation] Geometric Crystals, Elementary Characters and Monomial Realizations of Crystal Bases2011

    • Author(s)
      中島俊樹
    • Organizer
      International conference "Symmetries, Integrable Systems and Representations"
    • Place of Presentation
      Lyon, France(招待講演)
    • Year and Date
      2011-12-15
    • Related Report
      2011 Annual Research Report
  • [Presentation] Generalized Minors and Crystal Bases2011

    • Author(s)
      中島俊樹
    • Organizer
      研究会「可積分系とその周辺」
    • Place of Presentation
      大阪大学(招待講演)
    • Year and Date
      2011-07-02
    • Related Report
      2011 Annual Research Report
  • [Presentation] Decorated Geometric Crystals, Monomial Realizations and Polyhedral Realizations of Crystal Bases2010

    • Author(s)
      T.Nakashima
    • Organizer
      GCOE Conference : Algebraic and geometric aspects of discrete integrable systems-integrable systems and cluster algebras
    • Place of Presentation
      東京大学
    • Year and Date
      2010-12-15
    • Related Report
      2010 Annual Research Report
  • [Presentation] RSK type correspondcnce of Pictures and Littlewood-Richardson Crystals2010

    • Author(s)
      中島俊樹
    • Organizer
      2010年日本数学会秋季代数学分科会、無限可積分系セッション
    • Place of Presentation
      名古屋大学
    • Year and Date
      2010-09-25
    • Related Report
      2010 Annual Research Report
  • [Presentation] Geometric and Unipotent Bicrystals and Polyhedral Realization of Crystal bases2010

    • Author(s)
      T.Nakashima
    • Organizer
      ICM Satellite Conference, Algebraic and Combinatorial Approaches to Representation Theory
    • Place of Presentation
      National Institute of Advanced Studies, Bangalore, India
    • Year and Date
      2010-08-12
    • Related Report
      2010 Annual Research Report
  • [Presentation] 幾何結晶と多面体表示I,II2010

    • Author(s)
      T.Nakashima
    • Organizer
      表現論、組み合わせ論とその周辺
    • Place of Presentation
      琉球大学
    • Related Report
      2010 Annual Research Report
  • [Presentation] Elementary characters and monomial realizations of crystal bases

    • Author(s)
      中島 俊樹
    • Organizer
      Algebraic and Combinatorial approaches to representation
    • Place of Presentation
      University of California Riverside, USA
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Monomial realizations of crystal bases and elementary characters

    • Author(s)
      中島 俊樹
    • Organizer
      The XXIX International Colloquium on Group-Theoretic Methods in Physics
    • Place of Presentation
      Nankai University, China
    • Related Report
      2012 Annual Research Report
    • Invited
  • [Presentation] Decorated Geometric Crystals and Monomial Realizations of Crystals

    • Author(s)
      中島 俊樹
    • Organizer
      American Mathematical Society Fall SoutheasternSection Meeting,Special Session on Geometric andAlgebraic Aspects of Representation Theory
    • Place of Presentation
      Tulane University, New Orleans, USA
    • Related Report
      2012 Annual Research Report
    • Invited

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Published: 2010-08-23   Modified: 2019-07-29  

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