• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Construction of affine geometric crystals and Representation theory of crystal bases

Research Project

Project/Area Number 19540050
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  Sophia University, 理工学部, 教授 (60243193)

Co-Investigator(Kenkyū-buntansha) SHINODA Kenーichi  上智大学, 理工学部, 教授 (20053712)
TSUZUKI Masao  上智大学, 理工学部, 准教授 (80296946)
GOMI Yasushi  上智大学, 理工学部, 講師 (50276515)
Project Period (FY) 2007 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000)
Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2008: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2007: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords結晶基底 / 幾何結晶 / 超離散化 / ヘッケ環 / 代数群 / Tropical R写像 / 保型形式 / 熱帯化 / Tropical R map / マルコフトレース
Research Abstract

For most affine Kac-Moody algebras, we constructed geometric crystals associated with the simplest KR modules. We also obtained tropical R maps explicitly. Furthermore, only for sl (2)-case, certain universal type tropical R maps have been constructed. As a criterion for homogeneity of geometric crystals, we showed that the crystal obtained by the ultra-discretization is connected. By this criterion, it turns out that we can show easily the uniqueness of tropical R maps. We got epsilon systems of type A.

Report

(5 results)
  • 2009 Annual Research Report   Self-evaluation Report ( PDF )   Final Research Report ( PDF )
  • 2008 Annual Research Report
  • 2007 Annual Research Report
  • Research Products

    (29 results)

All 2010 2009 2008 2007 Other

All Journal Article (15 results) (of which Peer Reviewed: 15 results) Presentation (13 results) Remarks (1 results)

  • [Journal Article] Affine Geometric Crystal of type D^{(3)}_42010

    • Author(s)
      中島俊樹(M.Igarashi)
    • Journal Title

      Contemporary Mathematics 506

      Pages: 215-226

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Affine Geometric Crystal of type D^{(3)}_4.2010

    • Author(s)
      中島俊樹(M. Igarashi)
    • Journal Title

      Contemporary Mathematics 506

      Pages: 215-226

    • Related Report
      2009 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Universal tropical R map of sl (2) and prehomogeneous geometric crystals2009

    • Author(s)
      中島俊樹
    • Journal Title

      RIMS Kokyuroku Bessatsu B11

      Pages: 101-116

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Universal tropical R map of sl(2) and prehomogeneous geometric crystals.2009

    • Author(s)
      中島俊樹
    • Journal Title

      RIMS Kokyuroku Bessatsu B11

      Pages: 101-116

    • Related Report
      2009 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Universal tropical R map of sl(2)and prehomogeneous geometric crystals2009

    • Author(s)
      中島俊樹
    • Journal Title

      RIMS Kokyuroku Bessatsu B11

      Pages: 101-116

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Certain Rankin-Selberg integrals for unitary groups2009

    • Author(s)
      都築正男
    • Journal Title

      Tohoku Math.J. 61

      Pages: 115-164

    • NAID

      110007031057

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Spectral square means for period integrals of wave functions on real hyperbolic spaces2009

    • Author(s)
      都築正男
    • Journal Title

      J.Number Theory 129

      Pages: 1387-1438

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Affine Geometric Crystals and Limitof Perfect Crystals2008

    • Author(s)
      中島俊樹(M.Kashiwara M.Okado)
    • Journal Title

      Transactions in American Mathematical Society 30

      Pages: 3645-3687

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Affine Geometric Crystals and Limit of Perfect Crystals.2008

    • Author(s)
      中島俊樹, (M. Kashiwara, M. Okado)
    • Journal Title

      Transactions in American Mathematical Society 30

      Pages: 3645-3687

    • Related Report
      2009 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Affine Geometric Crystals and Limit of Perfect Crystals2008

    • Author(s)
      中島俊樹(with M. Kashiwara M. Okado)
    • Journal Title

      Transactions in American Mathematical Society 30

      Pages: 3645-3687

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Limit period formulas for a certain syrmmetric pair2008

    • Author(s)
      都築正男
    • Journal Title

      J. Funct. Anal. 255

      Pages: 1139-1190

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Affine Geometric Crystals and Limit of Perfect Crystals2008

    • Author(s)
      中島 俊樹
    • Journal Title

      Transactions of American Mathematical Society 360

      Pages: 3645-3686

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Affine Geometric Crystal of type G^{(1)}_22007

    • Author(s)
      中島俊樹
    • Journal Title

      Contemporary Mathematics 442

      Pages: 179-192

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Affine Geometric Crystal of type G^{(1)}_2.2007

    • Author(s)
      中島俊樹
    • Journal Title

      Contemporary Mathematics 442

      Pages: 179-192

    • Related Report
      2009 Self-evaluation Report
    • Peer Reviewed
  • [Journal Article] Affine Geometric Crystal of type G^{(1)}_22007

    • Author(s)
      中島 俊樹
    • Journal Title

      Contemporary Mathematics 442

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Presentation] Admissible Pictures and Littlewood-Richardson Crystals2009

    • Author(s)
      中島俊樹
    • Organizer
      International Workshop on Combinatorial and Geometric approach to Representation Theory
    • Place of Presentation
      Seoul National University, Korea
    • Year and Date
      2009-09-21
    • Related Report
      2009 Final Research Report
  • [Presentation] Admissible Pictures and Littlewood-Richardson Crystals.2009

    • Author(s)
      中島俊樹
    • Organizer
      International Workshop on Combinatorial and Geometric approach to Representation Theory
    • Place of Presentation
      Seoul National University, Korea
    • Year and Date
      2009-09-21
    • Related Report
      2009 Self-evaluation Report
  • [Presentation] Admissible Pictures and Littlewood-Richardson Crystals2009

    • Author(s)
      中島俊樹
    • Organizer
      International workshop on combinatorial and geometric approach to representation theory
    • Place of Presentation
      Seoul National University
    • Year and Date
      2009-09-21
    • Related Report
      2009 Annual Research Report
  • [Presentation] Epsilon Systems of geometric Crystals", Geometric Aspects of Discrete and Ultra-Discrete Integrable Systems2009

    • Author(s)
      中島俊樹
    • Organizer
      A satellite meeting of the Isaac Newton Institute Programme on Desicrete Integrable Systems
    • Place of Presentation
      The University of Glasgow, United Kingdom
    • Year and Date
      2009-03-31
    • Related Report
      2009 Final Research Report
  • [Presentation] Epsilon Systems of geometric Crystals.2009

    • Author(s)
      中島俊樹
    • Organizer
      Geometric Aspects of Discrete and Ultra-Discrete Integrable Systems, (A satellite meeting of the Isaac Newton Institute Programme on Desicrete Integrable Systems)
    • Place of Presentation
      The University of Glasgow, United Kingdom.
    • Year and Date
      2009-03-31
    • Related Report
      2009 Self-evaluation Report
  • [Presentation] Geometric Aspects of Discrete and Ultra-Discrete Integrable Systems2009

    • Author(s)
      中島俊樹
    • Organizer
      Epsilon Systems of geometric Crystals
    • Place of Presentation
      Glasgow, United Kingdam
    • Year and Date
      2009-03-31
    • Related Report
      2008 Annual Research Report
  • [Presentation] Geometric Crystals on flag varieties andmaximal unipotent subgroups2008

    • Author(s)
      中島俊樹
    • Organizer
      Workshop, 'Crystals and Tropical Combinatorics'
    • Place of Presentation
      Kansai Seminar House, 京都
    • Year and Date
      2008-08-30
    • Related Report
      2009 Final Research Report
  • [Presentation] Geometric Crystals on flag varieties andmaximal unipotent subgroups.2008

    • Author(s)
      中島俊樹
    • Organizer
      Workshop, 'Crystals and Tropical Combinatorics'
    • Place of Presentation
      Kansai Seminar House, 京都
    • Year and Date
      2008-08-30
    • Related Report
      2009 Self-evaluation Report
  • [Presentation] Affine Tropical R and Prehomogeneous Geometric Crystals2008

    • Author(s)
      中島俊樹
    • Organizer
      Quantum Affine Lie Algebras, Extended Affine Lie Algebras, and Applications
    • Place of Presentation
      Banff International Research Stations, Canada
    • Year and Date
      2008-03-05
    • Related Report
      2009 Final Research Report
  • [Presentation] Affine Tropical R and Prehomogeneous Geometric Crystals.2008

    • Author(s)
      中島俊樹
    • Organizer
      Quantum Affine Lie Algebras, Extended Affine Lie Algebras, and Applications
    • Place of Presentation
      Banff International Research Stations
    • Year and Date
      2008-03-05
    • Related Report
      2009 Self-evaluation Report
  • [Presentation] Affine Tropical R and Prehomogeneous Geometric Crystals2008

    • Author(s)
      中島 俊樹
    • Organizer
      Quantum Affine Lie Algebras, Extended Affine Lie Algebras, and Applications
    • Place of Presentation
      Banff Internationl Research Station,Canada
    • Year and Date
      2008-03-05
    • Related Report
      2007 Annual Research Report
  • [Presentation] Uniqueness of tropical R2007

    • Author(s)
      中島俊樹
    • Organizer
      組合せ論的表現論の拡がり
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2007-10-25
    • Related Report
      2009 Final Research Report
  • [Presentation] Uniqueness of tropical R'2007

    • Author(s)
      中島俊樹
    • Organizer
      組合せ論的表現論の拡がり
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2007-10-25
    • Related Report
      2009 Self-evaluation Report
  • [Remarks] ホームページ等

    • Related Report
      2009 Final Research Report

URL: 

Published: 2007-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi