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Recent development of special functins … an approach from the representation thery and the complex integrals

Research Project

Project/Area Number 15340003
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionTokyo Institute of Technoroly

Principal Investigator

MIMACHI Katsuhisa  Tokyo Institute of Technology, Grad.school of Sci. and Tech., Professor., 大学院理工学研究科, 教授 (40211594)

Co-Investigator(Kenkyū-buntansha) KUROKAWA Nobushige  Tokyo Institute of Technology, Grad.school of Sci. and Tech., Professor., 大学院理工学研究科, 教授 (70114866)
OCHIAI Hiroyuki  Nagoya Univ., Grad. school. of Math., Professor., 大学院多元数理科学研究科, 教授 (90214163)
TAKATA Toshie  Niigata Univ., Inst. of Sci. and Tech., Associate Professor., 自然科学系, 助教授 (40253398)
Project Period (FY) 2003 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥12,400,000 (Direct Cost: ¥12,400,000)
Fiscal Year 2006: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2005: ¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2004: ¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2003: ¥3,600,000 (Direct Cost: ¥3,600,000)
Keywordstwisted cycles / intersection numbers / Selberg type integral / conformal field theory / configuration space / Jones polynomials / zeta regularized products / resonant condition / 接続問題 / 一般超幾何函数 / モノドロミー / 共鳴状態 / 捩れサイクル / セルバーグ積分 / 表現論 / 代数解析 / ゼータ正規化 / 複素積分 / 交差数 / q超幾何級数 / 寺田模型
Research Abstract

Mimachi realized an irreducible representation of the Iwahori-Hecke algebra on the twisted homology group associated with a Selberg type integral. It was first constructed in the context of conformal field theory by Tsuchiya-Kanie. Our construction is based on the study of the homology group under a resonant condition on the exponents of integrands. We stress the importance of the study of integrals under such a resonant condition to the study of hypergeometric type functions and spherical functions. Mimachi with H. Ochiai (Nagoya) and M.Yoshida (Kyushu) formulated the concept of visible cycles and invisible cycles, and determined the dimension of the spaces of visible cycles under a resonant condition in some examples.
Mimachi with M.Yoshida calculated intersection numbers of twisted cycles associated with a Selberg type integral. It gives a natural interpretation of the coefficients of the four-point correlation function, in conformal field theory, calculated by Dotsenko-Fateev. This is an answer to a long standing problem of clarifying the meaning of such coefficients appearing in correlation functions. In higher dimensional cases, the Terada model (nonsingular model arising from the point configuration) plays an important role.
Kurokawa with M.Wakayama (Kyushu) studied generalized zeta regularizations. It shows that a discrete version of intersection numbers of twisted cycles should be settled.
Takata studied a q-hypergeometric series which appears as a factor of the n-colored Jones polynomial associated with a twisted knot or a torus knot and derived the A-polynomials associated with them.

Report

(5 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (33 results)

All 2006 2005 2004 2003 Other

All Journal Article (26 results) Book (1 results) Publications (6 results)

  • [Journal Article] Milnor's multiple gamma functions2006

    • Author(s)
      N.Kurokawa, H.Ochiai, M.Wakayama
    • Journal Title

      Jour. Ramanujan Math. Soc. 21

      Pages: 153-167

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Homological Representations of the Iwahori-Hecke algebra associated with a Selberg-type integral2005

    • Author(s)
      K.Mimachi
    • Journal Title

      Intern. Math. Res. Notices 33

      Pages: 2031-2057

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Homological Representations of the Iwahori-Hecke algebra associated with a Selberg-type integral2005

    • Author(s)
      K.Mimachi
    • Journal Title

      Intern.Math.Res.Notices 33

      Pages: 2031-2057

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A generalization of the Beta integral arising from the Knizhnik-Zamolodchikov equation for the vector representations of types B_n, C_n and D_n2005

    • Author(s)
      K.Mimachi, T.Takamuki
    • Journal Title

      Kyushu J.Math. 59

      Pages: 117-126

    • NAID

      110001263495

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Homological representations of the Iwahori-Hecke algebra associated with a Selberg type integral2005

    • Author(s)
      K.Mimachi
    • Journal Title

      International Mathematics Research Notices 33

      Pages: 2031-2057

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Differential algebraicity of multiple sine functions,2005

    • Author(s)
      N.Kurokawa, M.Wakayama
    • Journal Title

      Letters in Math.Phys. 71

      Pages: 75-82

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A generalization of the beta integral arising from the Knizhnik-Zamolodchikov equation for the vector representations of types B n, C n and D n2005

    • Author(s)
      K.Mimachi, T.Takamuki
    • Journal Title

      Kyushu J.of Math. 59

      Pages: 117-126

    • NAID

      110001263495

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Intersection numbers for loaded cycles associated with Selberg-type integrals2004

    • Author(s)
      K.Mimachi, K.Ohara, M.Yoshida
    • Journal Title

      Tohoku Math. J. 56

      Pages: 531-551

    • NAID

      110001043908

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Intersection numbers of twisted cycles associated with the Selberg integral and an application to the conformal field theory2004

    • Author(s)
      K.Mimachi, M.Yoshida
    • Journal Title

      Commun. Math. Phys. 250

      Pages: 23-45

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Reshetikhin Turaev invariants of Seifert 3-monifolds for classical simple Lie algebras2004

    • Author(s)
      H.S.Kold, T.Takata
    • Journal Title

      J. Knot Theory and Its Ramification 13

      Pages: 617-668

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Zeta regularized product expansion for multiple trigonometric functions2004

    • Author(s)
      N.Kurokawa, M.Wakayama
    • Journal Title

      Tokyo J. Math. 27

      Pages: 469-480

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Intersection numbers for loaded cycles associated with Selberg-type integrals2004

    • Author(s)
      K.Mimachi, K.Ohara, M.Yoshida
    • Journal Title

      Tohoku Math.J. 56

      Pages: 531-551

    • NAID

      110001043908

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Intersection numbers of twisted cycles associated with the Selberg integral and an application to the conformal field theory2004

    • Author(s)
      K.Mimachi, M.Yoshida
    • Journal Title

      Commun.Math.Phys. 250

      Pages: 23-45

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Reshetikhin Turaev invariants of Seifert 3-monifolds for classical simple Lie algebras2004

    • Author(s)
      H.S.Kold, T.Takata
    • Journal Title

      J.Knot Theory and Its Ramif. 13

      Pages: 617-668

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Zeta regularized product expansion for multiple trigonometric functions2004

    • Author(s)
      N.Kurokawa, M.Wakayama
    • Journal Title

      Tokyo J.Math. 27

      Pages: 469-480

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Zeta regularizations2004

    • Author(s)
      N.Kurokawa, M.Wakayama
    • Journal Title

      Acta Applicandae Mathematicae 81

      Pages: 147-166

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Zeta reguralized product expression for multiple trigonometric functions2004

    • Author(s)
      N.Kurokawa, M.Wakayama
    • Journal Title

      Tokyo J.Math. 27

      Pages: 469-480

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Reshetikhin-Turaev invariants of Seifert-manifolds for classical simple Lie algebras2004

    • Author(s)
      H.S.Kold, T.Takata
    • Journal Title

      J.Knot Theory and Its Ramifications 13

      Pages: 617-668

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Intersection theory for loaded cycles IV-resonant cases2003

    • Author(s)
      K.Mimachi, H.Ochiai, M.Yoshida
    • Journal Title

      Math. Nach. 260

      Pages: 67-77

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Intersection theory for loaded cycles IV-resonant cases2003

    • Author(s)
      K.Mimachi, H.Ochiai, M.Yoshida
    • Journal Title

      Math.Nach. 260

      Pages: 67-77

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Connection matrices associated with the generalized hypergeometric function _3F_2

    • Author(s)
      K.Mimachi
    • Journal Title

      Funkt. Ekvac. (to appear)印刷中

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Regularizable cycles associated with a Selberg-type integral under some resonance condition

    • Author(s)
      K.Mimachi, M.Yoshida
    • Journal Title

      Internat. Jour. Math. (to appear)印刷中

    • Related Report
      2006 Annual Research Report
  • [Journal Article] A formula for the colored Jones polynomial of 2-bridge knots

    • Author(s)
      T.Takata
    • Journal Title

      Kyungpook Math. Jour. (to appear)印刷中

    • Related Report
      2006 Annual Research Report
  • [Journal Article] A special value of the spectral zeta function of the non-commutative harmonic oscillators

    • Author(s)
      H.Ochiai
    • Journal Title

      Ramanujan Journal (掲載予定)(決定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Invariant distributions on a non-isotropic pseudo-Riemannian symmetric space of rank one

    • Author(s)
      H.Ochiai
    • Journal Title

      Indag.Math. (掲載予定)(決定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Theta lifting of Nilpotent orbits for symmetric pairs

    • Author(s)
      K.Nishiyama, H.Ochiai, C.B.Zhu
    • Journal Title

      Trans.of A.M.S. (掲載予定)(決定)

    • Related Report
      2005 Annual Research Report
  • [Book] 群論の進化2004

    • Author(s)
      堀田良之, 渡辺敬一, 庄司俊明, 三町勝久
    • Total Pages
      445
    • Publisher
      朝倉書店
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2004 Annual Research Report
  • [Publications] K.Mimachi, H.Ochiai, M.Yoshida: "Intersection theory for loaded cycles IV - resonant cases"Match.Nachr.. 260. 67-77 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Mimachi, M.Yoshida: "The reciplocity relation of the Selberg function"J.Comput.and Appl.Math.. 160. 209-215 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Mimachi, K.Ohara, M.Yoshida: "Intersection numbers for loaded cycles associated with Selberg type integrals"Tohoku Math.J.. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Mimachi, M.Yoshida: "Intersection numbers of twisted cycles associated with the Selberg integral and an application to the conformal field theory"Commun.Match.Phys.. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] N.Kurokawa, M.Wakayama: "Zeta regularizations"Acta Applicandae Mathematicae. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Takata: "Reshetikhin-Turaev invariants of Seifert 3-manifolds for classical simple Lie algebras"J.Knot Theory and Its Ramifications. (to appear).

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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