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Algebraic Analysis of Representation Theory

Research Project

Project/Area Number 13440006
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

KASHIWARA Masaki  KYOTO UNIVERSITY, Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (60027381)

Co-Investigator(Kenkyū-buntansha) MIWA Tetsuji  KYOTO UNIVERSITY, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10027386)
NAKAJIMA Hiraku  KYOTO UNIVERSITY, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00201666)
TANIZAKI Toshiyuki  Osaka City University, Faculty of Science, Professor, 大学院・理学研究科, 教授 (70142916)
内藤 聡  筑波大学, 数学系, 助教授 (60252160)
NAKASHIMA Toshiki  Sophia University, Faculty of Science and Technology, Associate Professor, 理工学部, 助教授 (60243193)
ANATOL Kirillov  KYOTO UNIVERSITY, Research Institute for Mathematical Sciences, Associate Professor (30346035)
兼田 正治  大阪市立大学, 理学部, 教授 (60204575)
Project Period (FY) 2001 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥16,700,000 (Direct Cost: ¥16,700,000)
Fiscal Year 2004: ¥4,100,000 (Direct Cost: ¥4,100,000)
Fiscal Year 2003: ¥4,100,000 (Direct Cost: ¥4,100,000)
Fiscal Year 2002: ¥4,100,000 (Direct Cost: ¥4,100,000)
Fiscal Year 2001: ¥4,400,000 (Direct Cost: ¥4,400,000)
Keywordsrepresentation theory / crystal bases / quantum group / D-modules / 幾何学的表現論 / 可解模型 / 組み合わせ
Research Abstract

In this project, we focused at geometric and combinatorial aspects of representation theory. Here are main achievements in these four years.
1.(1)In the coarse of the study of the form factors of exactly solvable models as integrals, their integrands have a symmetry of affine quantum groups (Miwa et al.) The representation thus obtained is in fact the tensor product of integrable representations in positive level and negative level. This result will be proved by using the result by Nakajima given below. (2)Nakajima studied global bases and crystal bases of affine quantum groups. In particular, he showed the global bases with extremal weight correspond one-to-one to the irreducible representation of the general linear groups.
2.Schapira constructed a canonical stack on the symplectic manifolds. It contains a parameter, and when it vanishes, the stack is equivalent to the one of modules over the ring of the functions. 3.Tanisaki achieved the one-to-one correspondence between D-modules on the quantized flag manifold and the modules over the quantum group. 4.Nakashima constructed geometric crystals associated to the Schubert cells and prove that their ultra-discretization coincide with the crystal for Demazure modules. Kashiwara and Nakashima, together with Okado, is studying the method to use another flag manifold in order to obtai the perfect crystals.

Report

(5 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (42 results)

All 2005 2004 2003 Other

All Journal Article (22 results) Publications (20 results)

  • [Journal Article] Level zero fundamental representations over quantized affine algebras and demazure modules2005

    • Author(s)
      Masaki, Kashiwara
    • Journal Title

      Publ.RIMS, Kyoto Univ. 41

      Pages: 223-250

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Crystar bases for quantum generalized Kac-Moody algebras2005

    • Author(s)
      Kyeonghoon Jeong, Seok-Jin Kang, Masaki Kashiwara
    • Journal Title

      Proc.London Math.Soc. 90

      Pages: 395-438

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The Beilinson-Bernstein correspondence for quantized enveloping algebras2005

    • Author(s)
      Toshiyuki Tanisaki
    • Journal Title

      Math.Z. (in press)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Geometric crystals on Schubert varieties2005

    • Author(s)
      Toshiki 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
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Crystal bases for quantum generalized Kac-Moody algebras2005

    • Author(s)
      Kyeonghoon Jeong, Seok-Jin Kang, Masaki Kashiwara
    • Journal Title

      Proc.London Math.Soc. 90

      Pages: 395-438

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Level zero fundamental representations over quantized affine algebras and Demazure modules2005

    • Author(s)
      Masaki Kashiwara
    • Journal Title

      Publ.RIMS, Kyoto Univ. 41

      Pages: 223-250

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The Beilinson-Bernstein correspondence for quantized enveloping algebras2005

    • Author(s)
      Toshiyuki Tanisaki
    • Journal Title

      Math.Z. (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Crystal bases for quantum generalized Kac-Moody algebras2005

    • Author(s)
      Kyeonghoon Jeong, Seok-Jin Kang, Masaki Kashiwara
    • Journal Title

      Proc. London Math. Soc. 90

      Pages: 395-438

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Level zero fundamental representations over quantized affine algebras and Demazure modules2005

    • Author(s)
      Masaki Kashiwara
    • Journal Title

      Publ. RIMS, Kyoto Univ. 41

      Pages: 223-250

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Form factors and action of U_<√<-1>>(sl^^〜_2) on ∞-Cycles2004

    • Author(s)
      M.Jimbo, T.Miwa, E.Mukhin, Y.Takeyama
    • Journal Title

      Commun.Math.Phys. 245

      Pages: 551-576

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Crystal bases and two-sided cells of quantum affine algebras2004

    • Author(s)
      Jonathan Beck, Hiraku Nakajima
    • Journal Title

      Duke Mathmatical Journal 123

      Pages: 335-402

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The represenation type of Hecke algebras of type B2004

    • Author(s)
      Susumu Ariki, Andrew Mathas
    • Journal Title

      Asvances in Mathamatics 181

      Pages: 134-159

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] On Kashiwara's equivalence in positive characteristic2004

    • Author(s)
      Masaharu Kaneda
    • Journal Title

      manuscripta math. 114

      Pages: 457-468

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Form factors and action of U_<√<-1>>(<sl>^^^〜_2)on ∞-Cycles2004

    • Author(s)
      M.Jimbo, T.Miwa, E.Mukhin, Y.Takeyama
    • Journal Title

      Commun.Math.Phys. 245

      Pages: 551-576

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Spaces of coinvariants and fusion product, I : from equivalence theorem to Kostka polynomials2004

    • Author(s)
      B.Feigin, M.Jimbo, R.Kedem, S.Loktev, T.Miwa
    • Journal Title

      Duke Mathematical Journal 125

      Pages: 549-588

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Crystal bases and two-sided cells of quantum affine algebras2004

    • Author(s)
      Jonathan Beck, Hiraku Nakajima
    • Journal Title

      Duke Mathematical Journal 123

      Pages: 335-402

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] An invitation to the generalized saturation conjecture2004

    • Author(s)
      Anatol N.Kirillov
    • Journal Title

      Publ.RIMS, Kyoto Univ. 40

      Pages: 1147-1239

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] The representation type of Hecke algebras of type B2004

    • Author(s)
      Susumu Ariki, Andrew Mathas
    • Journal Title

      Advances in Mathematics 181

      Pages: 134-159

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Annual Research Report 2004 Final Research Report Summary
  • [Journal Article] Form factors and action of U_<√<-1>>(<sI_2>^^^~) on ∞ -Cycles2004

    • Author(s)
      M.Jimbo, T.Miwa, E.Mukhin, Y.Takeyama
    • Journal Title

      Commun. Math. Phys. 245

      Pages: 551-576

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Extremal projectors of q-boson algebras2004

    • Author(s)
      Toshiki Nakashima
    • Journal Title

      Commun.Math.Phys. 244

      Pages: 285-296

    • Related Report
      2004 Annual Research Report
  • [Journal Article] On certain maximal cyclic modules for the quantized special linear algebra at a root of unity2003

    • Author(s)
      Masaharu Kaneda, Toshiki Nakashima
    • Journal Title

      Pacific Journal of Mathmatics 211

      Pages: 273-282

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Path model for a level-zero extremal weight module over a quantum affine algebra2003

    • Author(s)
      Satoshi Naito, Daisuke Sagaki
    • Journal Title

      IMRN 32

      Pages: 1731-1754

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] With Terasa Monteiro Fernandes, Pierre Schapira: "Truncated microsupport and holomorphic solutions of Modules"Ann.Scient.Ec.Norm.Sup.. 4serie, 36. 583-599 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Masaki Kashiwara: "Realizations of Crystals, Combinatorial and geometric representation theory : an international conference on combinatorial and geometric representation theory"Contemporary Mathematics. 325, AMS. 133-139 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Naito, D.Sagaki: "Three kinds of extremal weight vectors fixed by a diagram automorphism"J. algebra. Vol.268 no.1. 343-365 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Nakashima (with M.Kaneda): "On certain maximal cyclic modules for the quantized special linear algebras at root of Unity"Pacific Journal of Mathematics. 211, no.2. 273-282 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Nakajima (with Jonathan Beck): "t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras"Represent.Theory (elect.). 7. 259-274 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Takedshi Suzuki: "Classification of simple modules over degenerate double affine Hecke algebras of type A"Inter.Math.Res.Notices. Number43. 2313-2339 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Kashiwara, M.: "On Level Zero Representations of Quantized Affine Algebras"Duke Math. J.. 112・1. 117-175 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Miwa, T., Feigin, B., Kedem, R., Loktev, S., Mukhin, E.: "Combinatorics of the Sl_2 spaces of coinvariants : loop Heisenberg modules and recursion"Selecta Math. (N.S.). 8・3. 419-474 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Nakajima, H.: "Quiver varieties and t-analogs of q-characters of quantum affine algebras"Ann. of Math., to appear. (発行予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Kashiwara, M., Tanisaki, T.: "Parabolic Kazhdan-Lusztig polynomials and Schubert varietie"J. Algebra. 2492・2. 2401-2430 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Kirillov, Anatol N., Shimozono, M.: "A generalization of the Kostka-Foulkes polynomials"J. Algebraic Combin.. 15. 27-69 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Nakashima, T.: "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
      2002 Annual Research Report
  • [Publications] 谷崎俊之: "リー代数と量子群"共立出版. 267 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Masaki KASHIWARA: "Duality of D-modules on flag manifolds"International Mathematics Research Notices. 23. 491-509 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] Toshiyuki TANISAKI: "Parabolic Kazhdan-Lusztig polynominals and Schubert varieties"J.Algebra.

    • Related Report
      2001 Annual Research Report
  • [Publications] Masaharu KNEDA(with H.H.Anderson): "Filtrations on G_1T-modules"Proc.London Math.Soc.. (3)82. 614-616 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hiraku NAKAJIMA: "t-Analogue of the q-characters of finite dimensional representation of quantum affine algebras,in"Physics and Combinatorics""Proceeding of the Nagoya 2000 International Workshop,World Scientific. 195-218 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Satoshi NAITO: "On the Harish-Chandra homorphism for generalized kac-Moody algebra"Comm.Algebra. Vol.245No.3. 1069-1084 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Atsushi NAKAYASHI: "On the cohomologies of theta divisors of hyperelliptic Jacobians"Proceeding of the 9-th MSJ-IRI Tokyo 2000,M.Guest et al.ed..

    • Related Report
      2001 Annual Research Report
  • [Publications] 柏原正樹: "代数解析概論、岩波講座、現代数学の展開9"岩波書店. 262 (2000)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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