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2006 Fiscal Year Final Research Report Summary

Piecewise Linear Representation Theory of Quantum Groups and Geometric Crystals

Research Project

Project/Area Number 16540039
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, Science and Thechnology, Professor (60243193)

Co-Investigator(Kenkyū-buntansha) SHINODA Ken-ichi  Sophia University, Science and Thechnology, Professor (20053712)
GOMI Yasushi  Sophia University, Science and Thechnology, Lecturer (50276515)
Project Period (FY) 2004 – 2006
KeywordsQuantum Grouos / Crystal Bses / Geometric Crystals / Tropicalization / Ultra-Discretizations / Algebraic Groups / Hecke Akgebras / Markov Trace
Research Abstract

The theory of geometric crystal is obtained as an analogus theory on algebraic varieties to crystal bases by considering certain group actions on the vatirety, which turns out to be an analogus operator to the crystal operators. It is well-known that by the tropicalization / ultra-discretization procedure we obtain crystals from geometric crystals.
The purpose of the research is to construct a geometric crystal structure on various algebraic varieties. In fact, I have succeeded to construct the geometric crystal structure on the Schubert varieties associated with Kac-Moody groups. Furthermore, I have constructed the geometric crystal structures on the affine Schubert varieties which is obtained from the translations of the extended Weyl groups. This geometric crystal possesses the following remarkable properties : it has a natural positive structure and the associated crystal is isomorphic to the so-called the limit of perfect crystals. We obtained the tropical R maps on the product of … More these geometric crystals, which is an analogus object to R-matrices. Perfect crystals are crucial objects in the study of affine type crystals and they play a central role in the theory of solvable lattice models in mathematical physics and the theory of Kirillov-Reshetikhin modules.
As for the representation of affine quantum groups t roots of 1, we obtain the sufficient and necessary condition for that two evaluation representations are isomorphic to each other.
The Markov trace is one of important topological invariants. From the view point of topology and representation theory, the Markov trace turns out to be an rather interesting object which we should study. In order to treat the Markov trace in the general framework, Gomi defined the Markov property and present the way to construct the Markov trace by the unified method.
Shinoda succeeded in obtaining certain interesting results on relations between zeta functions and the Gel'fand Graev representations of finite reductive groups by performing the explicit calculations. Less

  • Research Products

    (8 results)

All 2006 2005

All Journal Article (6 results) (of which Peer Reviewed: 3 results) Presentation (2 results)

  • [Journal Article] Evaluation representations and quantum affine algebras at roots of unity2006

    • Author(s)
      Y.Abe and T.Nakashima
    • Journal Title

      Journal of Mathematical Physics 47

      Pages: 083514 1-28

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] The Markov traces and the Fourier transforms2006

    • Author(s)
      Y.Gomi
    • Journal Title

      Journal of Algebra 303

      Pages: 566-591

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Evaluation representations and quantum affine Algebras at roots of unity2006

    • Journal Title

      Journal of Mathematical Physics 47

      Pages: 1-28

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The Markov traces and the Fourier transforms2006

    • Author(s)
      Y., Gomi
    • Journal Title

      Journal of Algebra 303

      Pages: 566-591

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Geometric Crystals on Schubert Varieties2005

    • Author(s)
      T.Nakashima
    • Journal Title

      Journal of Geometry and Physics 53

      Pages: 197-225

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Geometric Crystals on Schubert Varieties2005

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

      Journal of Geometry and Physics 53

      Pages: 197-225

    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Affine Geometric Crystals and Tropical R2005

    • Author(s)
      T., Nakashima
    • Organizer
      Lie Algebras, Vertex Operator Algebras and Their Applications
    • Place of Presentation
      Norht Carolina State University, USA
    • Year and Date
      20050500
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Affine Geometric Crystals and Tropical R2005

    • Author(s)
      中島 俊樹
    • Organizer
      Lie Algebras, Vertex Operator Algebras and Their Applications
    • Place of Presentation
      North Carolina State Univ USA
    • Year and Date
      2005-05-17
    • Description
      「研究成果報告書概要(和文)」より

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Published: 2010-02-04  

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