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Combinatorial Representation Theory of Quantum Groups

Research Project

Project/Area Number 13640043
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, Department of Mathematics, Ass.Professor, 理工学部, 助教授 (60243193)

Co-Investigator(Kenkyū-buntansha) GOMI Yasushi  Sophia University, Department of Mathematics, Assistant, 理工学部, 助手 (50276515)
YOKONUMA Takeo  Sophia University, Department Mathematics, Professor, 理工学部, 教授 (00053645)
SHINODA Ken-ichi  Sophia University, Department Mathematics, Professor, 理工学部, 教授 (20053712)
古閑 義之  上智大学, 理工学部, 助手 (20338429)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2003: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2002: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2001: ¥1,300,000 (Direct Cost: ¥1,300,000)
KeywordsQuantum Groups / Crystal Bases / Braid Groups / Polyhedral Realizations / Maximal Cyclic Representations / coinvariant algebra / Hecke algebra / Gauss sum / 表現論 / 極外ベクトル / 極大巡回加群 / ゼータ関数 / イプシロン因子 / 組合せ論 / 1の巾根
Research Abstract

We studied representation theory of quantum groups and related combinatorics. In particular, the head investigator researched theory of crystal bases by the method of polyhedral realizations. Polyhedral realization is to embed a crystal base in certain infinite integer lattice and realized it as a lattice points in some convex polyhedron. Its feature is to enable us to calculate many things explicitly. Indeed, for several types of quantum groups, he succeeded in giving explicit form of crystal bases and extremal vectors. He also studied quantum groups at roots of unity. As for maximal cyclic representations of type A, specializing its parameters properly, he gave irreducible modules of restricted quantum algebra. Furthermore, he compared it with infinitesimal Verma modules.
K.Shinoda studied Chevalley groups and Hecke algebras. For Chevalley group G_2(q), we gave Gauss sums for modular representation of degree 7 and 11 unipotent irreducible representations. In the course of its proof, he also presented several kinds of summation formulae. In particular, he showed that on some elements, character values coincide with general Kloosterman sum.
Y.Gomi gave some presentations of pure braid groups associated with Coxeter groups by combinatorial methods, in particular, by Reidemeister-Schreier theorem

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report

Research Products

(30 results)

All Other

All Publications (30 results)

  • [Publications] T.Nakashima: "Polytopes for Crystallized Demazure Modules and Extremal Vectors"Communications in Algebra. 30,3. 1349-1367 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Nakashima: "Irreducible Modules of Finite Dimensional Quantum Algebras of type A at roots of unity"Journal of Mathematical Physics. 43,4. 2000-2014 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Nakashima, M.Kaneda: "On certain maximal cyclic modules for the quantized special linear algebras at a root of unity"Pacific Journal of Mathematics. 211,2. 273-282 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Nakashima: "Extremal Projectors of $q$-boson algebras"Communications in Mathematical Physics. 44. 285-296 (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Shinoda, N.Saito: "Some character sums and Gauss sums over $G_2(q)$"Tokyo Journal of Mathematics. 24. 277-289 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Shinoda, C.W.Curtis: "Zeta functions and functional equations associated with the components of the Gelfand-Graev representations of a finite reductive group"Advanced Studies in Pure Mathematics. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Gomi, F.Digne: "Presentation of pure braid groups"Journal of Knot Theory Ramifications. 10,4. 609-623 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Gomi, I.Nakamura, K.Shinoda: "Coinvariant Algebras of finite subgroups of SL(3,C)"Canadian Journal of Mathematics. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Nakashima: "Polytopes for Crystallized Demazure Modules and Extremal Vectors"Communications in Algebra. Vol.30,No.2. 1349-1357 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Nakashima: "Irreducible Modules of Finite Dimensional Quantum Algebras of type A at roots of unity"Journal of Mathematical Physics. Vol.43,No.4. 2000-2014 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Nakashima, M.Kaneda: "Irreducible Modules of Finite Dimensional Quantum Algebras of type A at roots of unity"Pacific Journal of Mathematics. Vol.211,No.2. 273-282 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Nakashima: "Extremal Projectors of $q$-boson algebras"Communications in Mathematical Physics. Vol.244. 285-296 (2004)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Shinoda, N.Saito: "Some character sums and Gauss sums over $g_2(q)$."Tokyo Journal of Mathematics. Vol.24. 277-289 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] K.Shinoda, S.W.Curtis: "Zeta functions and functional equations associated with the components of the Gelfand-Graev representations of a finite reductive group"Advanced Studies in Pure Mathematics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Gomi, F.Digne: "Presentation of pure braid groups"Journal of Knot Theory Ramifications. Vol.10,No.4. 503-523 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Y.Gomi, I.Nakamura, K.Shinoda: "Coinvariant Algebras of finite subgroups of SL(3,C)"Canadian Journal of Mathematics. (to Appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Nakashima: "Polytopes for Crystallized Demazure Modules and Extremal Vectors"Communications in Algebra. 30,3. 1349-1367 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Nakashima: "Irreducible Modules of Finite Dimensional Quantum Algebras of type A at roots of unity"Journal of Mathematical Physics. 43,4. 2000-2014 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Nakashima, M.Kaneda: "On certain maximal cyclic modules for the quantized special linear algebras at a root of unity"Pacific Journal of Mathematics. 211,2. 273-282 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Nakashima: "Extremal Projectors of $q$-boson algebras"Communications in Mathematical Physics. 244. 285-296 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Shinoda, N.Saito: "Some character sums and Gauss sums over $G_2(q)$"Tokyo Journal of Mathematics. 24. 277-289 (2001)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Shinoda, C.W.Curtis: "Zeta functions and functional equations associated with the components of the Gelfand-Graev representations of a finite reductive group"Advanced Studies in Pure Mathematics. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Gomi, F.Digne: "Presentation of pure braid groups"Journal of Knot Theory. 10,4. 609-623 (2001)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Gomi, I.Nakamura, K.Shinoda: "Coinvariant Algebras of finite subgroups of SL(3,C)"Canadian Journal of Mathematics. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] 中島俊樹: "Polytopes for Crystallized Demazure modules and Extremol Vectors"Communications in Algebra. 30.3. 1349-1367 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 中島俊樹: "Irreducible modules of finite dimensional Quantum algebras of type A at roots of unity"Journal of Mathematical Physics. 43.3. 2000-2014 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 中島俊樹: "On certain maximal cyclic modules for the quantized special linear algebras at a root of unity"Pacific Journal. (in press).

    • Related Report
      2002 Annual Research Report
  • [Publications] 筱田 健一: "Zeta functions and functional equations associated with the component of the Gelfand-Graev representations of a finite reductive group"Advanced Studies in Pure Mathematics. (in press).

    • Related Report
      2002 Annual Research Report
  • [Publications] 中島俊樹: "Polytopes for Crystallized Demazure Modules and Extremal vectors"Communications in Algebra. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] 中島俊樹: "Irreducible Modules of Finite Dimensional Quantum Algebras of type A at roots of unity"Journal of Mathematical Physics. (発表予定).

    • Related Report
      2001 Annual Research Report

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Published: 2001-03-31   Modified: 2016-04-21  

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