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Study of special functions based on representation and invariant theories

Research Project

Project/Area Number 11440043
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

UMEDA Toru  Math. Dept., Kyoto University, Associte Professor, 大学院・理学研究科, 助教授 (00176728)

Co-Investigator(Kenkyū-buntansha) NOUMI Masatoshi  Math. Dept., Kobe Univ., Prof., 大学院・自然科学研究科, 教授 (80164672)
MATSUZAWA Junichi  Dept. of technology, Kyoto University, Lecturer, 大学院・工学研究科, 講師 (00212217)
NOMURA Takaaki  Math. Dept., Kyoto University, Assoc. Prof., 大学院・理学研究科, 助教授 (30135511)
OEHIAI Hiroyuki  Math. Dept. Tokyo Inst. Technology, Assoc. Prof., 大学院・理学研究科, 助教授 (90214163)
WAKAYAMA Masato  Graduate School of Math. Kyushu Univ., Prof., 大学院・数理学研究院, 教授 (40201149)
菊地 克彦  京都大学, 大学院・理学研究科, 助手 (50283586)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥9,600,000 (Direct Cost: ¥9,600,000)
Fiscal Year 2001: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2000: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 1999: ¥4,000,000 (Direct Cost: ¥4,000,000)
Keywordsspecial functions / representation theory / invariant theory / invariant differential operations / Copelli identity / hypergeometric functions / determinant / Pfaphan / 群の表現 / 不変式 / 群行列式 / Lie環 / 不偏包絡環 / Poincare Birkhoff-Wittの定理 / 対称群 / 量子群 / リー環 / 普遍包絡環 / 球函数 / パーマネント / Wronski関係式 / 五角数定理
Research Abstract

The main object of the research is to find the group theoretical background behind the world of special functions and to utilize the symonetious for the special functions. Among them the theory of "dual pairs" is the key to our study, which explains many phenomina from the view-point of representation theory and the theory of invariants. We have Capilli type identities, now commtative harnomic oscillatws as the typical investizations where and pains work very well as the griding principle. On the other hand, for the hyprogeinctic from Rons and Pain lene transcendents, we have claified the grop gymmetric behind them. The helps a lot for deeper investigations of these fnctions.
As for the Capelli type identities, we got many interesting formlas including permanets and Pfuffians, not only for the determinants, Furthermore we found some Capelli type identity corresponding to the "group determinant". The invariant theoretic backgroud conneits these identities to some sphenicel functions. There are sort of unification of various objects.

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (35 results)

All Other

All Publications (35 results)

  • [Publications] T.Umeda: "On Turnbull identity for skew symmetric matrices"proc. Edinburgh Math. Soc.. 43. 379-393 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Itoh, T.Umeda: "On central elements in the universal enveloping algebra of the orthogonal Lie algebra"Compositio Math.. 127. 333-359 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Nomura: "On Penney's Cayley transform of a hemogeneous Siegel domain"J. Lie Theory. 11. 185-206 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y.Kajihara, M.Noumi: "Raising operators of row type for Macdonald polynomials"Compositio Math.. 120. 119-136 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] N.Kurokawa, M.Wakayama: "On ζ(3)"j. Ramanujan Math. Soc. 16. 205-214 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Ochiai: "Non-commutative harmonic oscillators and Fuchian indinary differential equations"Comm. Math. Phys.. 217. 357-373 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 梅田享, 黒川信重, 若山正人, 中島さち子: "ゼータの世界"日本評論社. 156 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Umeda: "On Turnbull identitfies for skew symmetric Matrics"Pior, Edinburgh Math. Soc. 43. 379-393 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Itoh, T.Umeda: "On central elements in the universal enveloping alegibra of the orthogonal Lie algebra"Conposition Math. 127. 333-359 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Nomura: "On Penny's Cayley transform of a homogeneous Segel domain"J. Lie Theory. 11. 185-206 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Y.Kajihara, M.Noumi: "Raising operetors of row-type for Macdonald polynomials"Corposiho Math. 120. 119-136 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] N.Karakawa, M.Wakayama: "On ζ(3)"J.Ramanujan Math.Soc. 16. 205-214 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] H.Ochiai: "Non-commutative harmonic osciltatris and Fuchiam ordinary differendial equatons"Comm.Math.Phys. 217. 357-373 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Umeda, N.Kurokawa, M.Wakayama, Sachiko Nakajima: "the world of zeta"Nihon-Hyoronsha(in Japanese). 156 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] M.Itoh, T.Uneda: "On central elements in the universal enveloping algebra of the orthogonal Lie algebra"Compositio Math.. 127. 333-359 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Nomura: "On Penney's Cayley transform of a homogeneous Siegel domain"J. Lie Theory. 11. 185-206 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Kajiwara, M.Noumi, Y.Yamada: "A Study on the fourth q-Painleve equation"J. Phys. A : Math. Gen. 34. 8563-8581 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] N.Kurokawa, M.Wakayama: "On ζ(3)"J. Ramanujan Math. Soc.. 16. 205-214 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] A.Parmeggiani, M.Wakayama: "Oscillator representations and systems of ordinary differential equations"Proc. Nat. Acad. Sci.. 98. 26-31 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Ochiai: "Non-commutative harmonic oscillators and Fuchian ordinary differential equations"Comm. Math. Phys.. 217. 357-373 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 黒川信重, 若山正人: "絶対カシミール元"岩波書店. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Umeda: "On Turnbull identity for skew-symmetric matrices"Proc.Edinburgh Math.Soc.. 43. 379-393 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Itoh and T.Umeda: "On central elements in the universal enveloping algebra of the orthogonal Lie algebras"Compositio Math.. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Nomura: "Invariant Berezin transforms"CRC Research Notes in Math.. 422. 19-40 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] 野海正俊,山田泰彦: "Painleve方程式の対称性"数学. 53. 62-75 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] N.Kurokawa,H.Kuroyama and M.Wakayama: "A formula for the multiplicity of the principal series in L^2 (Γ\G)"Forum Math.. 12. 757-766 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Ochiai: "Classification of commuting differential operators in two variables"Yang-Baxter systems, Non-linear models and their applications. 183-187 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 野海正俊: "パンルヴェ方程式-対称性からの入門-"朝倉書店. 201 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Toru Umeda: "On turnbull identity for skew-symmetric matrices"Proc. Edinburgh Math. Soc.. 43. 1-15 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Takaaki Nomura: "Invariant Berezin transforms"Proc. Conf. "Harmonic Analysis and Integral Geometry". (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] J.Matsuzawa and A.Omura: "Blow-ups of IP^2 and root system of type D"J. Math. Kyoto. Univ.. 39. 725-761 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Noumi and Y.Yamada: "Symmetries in the fourth Painleve equations and Odamoto polynomials"Nagoya Math. J.. 153. 53-86 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Ishikawa and M.Wakayama: "Applications of minor summation formulas II"J. Comb. Theo. Ser A. 88. 136-157 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Hiroyuki Ochiai: "A p-adic property of Taylor series of exp(χ+χ^p/p)"Hokkaido Math. J.. 28. 71-85 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 梅田亨・黒川信重・若山正人・中島さち子: "ゼータの世界"日本評論社. 156 (1999)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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