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Research on the holonomic q difference systems associated with the Jackson integrals of Weyl group invariant

Research Project

Project/Area Number 17540037
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionAoyama Gakuin University

Principal Investigator

ITO Masahiko  Aoyama Gakuin University, College of Science and Engineering, Associate Professor (30348461)

Co-Investigator(Kenkyū-buntansha) TANIGUCHI Kenji  Aoyama Gakuin University, College of Science and Engineering, Associate Professor (20306492)
KOIKE Kazuhiko  Aoyama Gakuin University, College of Science and Engineering, Professor (70146306)
SATSUMA Junkichi  Aoyama Gakuin University, College of Science and Engineering, Professor (70093242)
YANO Kouichi  Aoyama Gakuin University, College of Science and Engineering, Professor (60114691)
川村 友美  青山学院大学, 理工学部, 助手 (40348462)
木村 勇  青山学院大学, 理工学部, 助手 (40082820)
Project Period (FY) 2005 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,570,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥270,000)
Fiscal Year 2007: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2005: ¥1,500,000 (Direct Cost: ¥1,500,000)
KeywordsJackson integrals / Weyl group / q-difference system / basic hypergeometric series / connection formula / Vandermonde-type determinant / invariant polynomials of type F4 / ルート系 / Kazhdan-Lusztig多項式 / 複素簡約型リー群 / 対称式 / ホロノミックq-差分方程式 / カロジェロ模型
Research Abstract

A key reason to consider the Jackson integrals, which admit Weyl group symmetry, is to provide an explanation and an extension of a series of classical basic hypergeometric series and their q-difference equations with respect to their parameters.
In order to show the linear independence of the fundamental solutions of the q-difference system, the non-degeneracy of the Wronskian should be proved. For this purpose, Ito and K. Aomoto (Kyoto sangyo university) showed that the Wronskian is expressed as a product of q-gamma functions. Ito and Koike showed that a Vandermonde-type determinant, whose entries are the irreducible characters of the classical groups, is expressed as a power of the difference product. Technically speaking, the proof of the non-degeneracy of the Wronskian is obtained from the Vandermonde-type determinant, which is a limiting case of q-0 for the Wronskian. The formula given by Ito and Koike plays an important role in the result by Ito and Aomoto.
Ito and Y. Sanada (Tsuda collage) gave an explicit connection formula, which indicates that the general solution of the q-difference system of the BC1-type Jackson integral is expressed as a linear combination of the fundamental solutions of the system. They showed that the connection formula is equivalent to the classical hypergeometric transformation formula, which was discovered in 1950s by Sears and Slater. Consequently they found the Weyl group symmetry in the transformation formula and gave a very simple proof for the formula. Ito also gave an explicit connection formula for the BCn-type Jackson integral by extending it from the result of Ito and Sanada.
Taniguchi gave an elementary construction for the invariant polynomials of type F4 under the action of its Weyl group, using the degree 2 invariant polynomials under the action of Weyl group of type D4, based on the fact that Weyl group of type D4 is included in a normal subgroup of Weyl group of type F4.

Report

(4 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • Research Products

    (47 results)

All 2008 2007 2006 2005 Other

All Journal Article (31 results) (of which Peer Reviewed: 11 results) Presentation (16 results)

  • [Journal Article] On the Sears-Slater basic hypergeometric transformations2008

    • Author(s)
      M.Ito & Y.Sanada
    • Journal Title

      Ramanujan Journal (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] A multiple generalization of Slater's transformation formula for a very-well-poised-balanced_<2r>Φ_<2r>series2008

    • Author(s)
      Masahiko Ito
    • Journal Title

      The Quarterly Journal of Mathematics (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] BCn-type Jackson integral generalized from Gustafson's Cn-type sum2008

    • Author(s)
      K.Aomoto & M.Ito
    • Journal Title

      Journal of Difference Equations and Applications (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Structure of Jackson integrals of BCn type2008

    • Author(s)
      K.Aomoto & M.Ito
    • Journal Title

      Tokyo Journal of Mathematics (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] A multiple generalization of Slater's transformation formula for a very-well-Poised-balanced _<2r>φ_<2r> series2008

    • Author(s)
      Masahiko Ito
    • Journal Title

      The Quarterly Journal of Mathematicsに掲載受理

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] BCn-type Jackson integral generalized from Gustafson's Cn-typesum2008

    • Author(s)
      K.Aomoto & M.Ito
    • Journal Title

      Journal of Difference Equations and Applicationsに掲載受理

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] F4型ワイル群不変式の初等的構成法2006

    • Author(s)
      谷口 健二
    • Journal Title

      数理解析研究所講究録 1508

      Pages: 119-124

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] BCn型ジャクソン積分と墓本'対称式2006

    • Author(s)
      伊藤 雅彦
    • Journal Title

      数理解析研究所講究録 1497

      Pages: 79-87

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] A generalization of Weyl's denominator formulas for the classical groups2006

    • Author(s)
      M.Ito & K.Koike
    • Journal Title

      Journal of Algebra 302

      Pages: 817-825

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Askey-Wilson type integrals associated with root systems2006

    • Author(s)
      M.Ito
    • Journal Title

      Ramanujan Journal 12

      Pages: 131-151

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Another proof of Gustafson's Cn-type summation formula via elementary'symmetric polynomials2006

    • Author(s)
      M.Ito
    • Journal Title

      Publications of RIMS 42

      Pages: 523-549

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] q-difference shift for a BCn type Jackson integral arising from 'elementary'symmetric polynomials2006

    • Author(s)
      M.Itc
    • Journal Title

      Advances in Mathematics 204

      Pages: 619-646

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] An elementary construction of the invariant polynomials under the action of Weyl group of type F42006

    • Author(s)
      Kenji, Taniguchi
    • Journal Title

      RIMS Kokyuroku 1508

      Pages: 119-124

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Jackson integrals of type BCn and `'elementary' symmetric functions (in Japanese)2006

    • Author(s)
      Masahiko, Ito
    • Journal Title

      RIMS Kokyuroku 1497

      Pages: 79-97

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] A generalization of Weyl's denominator formulas for the classical groups2006

    • Author(s)
      Kazuhiko, Koike, Masahiko, Ito
    • Journal Title

      Journal of Algebra 302

      Pages: 817-825

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Askey-Wilson type integrals associated with root systems2006

    • Author(s)
      Masahiko, Ito
    • Journal Title

      Ramanujan Journal 12

      Pages: 131-151

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Another proof of Gustafson's Cn-type summation formula via 'elementary' symmetric polynomials2006

    • Author(s)
      Masahiko, Ito
    • Journal Title

      Publications of RIMS 42

      Pages: 523-549

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] q-difference shift for a BCn type Jackson integral from 'elementary' symmetric polynomials2006

    • Author(s)
      Masahiko, Ito
    • Journal Title

      Adv. in Math 204

      Pages: 19-646

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] q-difference shift for a BCn type Jackson integral arising from 'elementary'symmetric polynomials2006

    • Author(s)
      M.Ito
    • Journal Title

      Advances in Mathematics 204

      Pages: 619-646

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Another proof of Gustafson's Cn-type summation formula via 'elementary'symmetric polynomials2006

    • Author(s)
      M.Ito
    • Journal Title

      pulications of RIMS 42

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Askey-Wilson type integrals associated with systems2006

    • Author(s)
      M.ito
    • Journal Title

      Ramanujan Journal 12

    • Related Report
      2006 Annual Research Report
  • [Journal Article] A generalization of Weyl's denominator formulas for the classical groups2006

    • Author(s)
      M.Ito, K.Koike
    • Journal Title

      Journal of Algebra 302

    • Related Report
      2006 Annual Research Report
  • [Journal Article] BCn型ジャクソン積分と'基本'対称式2006

    • Author(s)
      伊藤雅彦
    • Journal Title

      数理解析研究所講究録 1497

      Pages: 79-87

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On the structure of Jackson integrals of B_n type and Holonomic q-difference equations2005

    • Author(s)
      K.Aomoto & M.Ito
    • Journal Title

      Proceedings of the Japan Academy 81,Ser.A,No.9

      Pages: 145-150

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On the structure of Jackson integrals of BCn Type and holonomic q-difference equations2005

    • Author(s)
      K., Aomoto, M., Ito
    • Journal Title

      Proc. Japan Acad. Ser. A Math., Sci. 81

      Pages: 145-150

    • NAID

      40007140160

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On the structure of Jackson integrals of $BC_n$ type and Holonomic q-difference equations2005

    • Author(s)
      K.Aomoto, M.Ito
    • Journal Title

      Proceedings of the Japan Academy 81, Ser. A, No.9

      Pages: 145-150

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the Sears-Slater basic hypergeometric transformations

    • Author(s)
      Masahiko, Ito, Yukari, Sanada
    • Journal Title

      Ramanujan Journal (in press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] A multiple generalization of Slater's transformation formula for a very-well-poised-balanced 2rΨ2r series

    • Author(s)
      Masahiko, Ito
    • Journal Title

      Quarterly Journal of Math (in press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] BCn-type Jackson integral generalized from Gustafson's Cn-type sum

    • Author(s)
      K., Aomoto, M., Ito
    • Journal Title

      Journal of Difference Equations and Applications (in press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Structure of Jackson integrals of BCn type

    • Author(s)
      K., Aomoto, M., Ito
    • Journal Title

      Tokyo Journal of Mathematics (in press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On the Sears-Slater basic hypergeometric transformations

    • Author(s)
      M.Ito, Y.Sanada
    • Journal Title

      Ramanujan Journal に掲載受理

    • Related Report
      2006 Annual Research Report
  • [Presentation] On the Sears-Slater basic hypergeometric transformations2008

    • Author(s)
      Masahiko Ito
    • Organizer
      Conference on Partitions, q-Series and Modular Forms
    • Place of Presentation
      University of Florida アメリカ
    • Year and Date
      2008-03-16
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] On the Sears-Slater basic hypergeometric transformations2008

    • Author(s)
      Masahiko, Ito
    • Organizer
      Conference on Partitions, q-Series and Modular Forms
    • Place of Presentation
      Univ. of Florida, U. S. A
    • Year and Date
      2008-03-16
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Sears-Slaterのvery-well-poised q-超幾何変換公式の多重級数への一拡張2007

    • Author(s)
      伊藤 雅彦
    • Organizer
      日本数学会年会
    • Place of Presentation
      埼玉大学
    • Year and Date
      2007-03-29
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On the Sears-Slater basic hypergeometric transformations2007

    • Author(s)
      Masahiko, Ito
    • Organizer
      the annual meeting of MSJ
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] ワイル群対称性によるvery-well-poised q-超幾何級数の変換公式の証明2006

    • Author(s)
      伊藤 雅彦, 真田 ゆかり
    • Organizer
      日本数学会秋季分科会
    • Place of Presentation
      大阪市立大学
    • Year and Date
      2006-09-22
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] G2型マクドナルド対称多項式のピエリ公式2006

    • Author(s)
      伊藤 雅彦
    • Organizer
      日本数学会年会
    • Place of Presentation
      中央大学
    • Year and Date
      2006-03-29
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Pieri formulas for the symmetric polynomials of type G22006

    • Author(s)
      Masahiko, Ito
    • Organizer
      the annual meeting of MSJ
    • Place of Presentation
      Chuo University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] a proof of transformation formula for a very-well-poised q-hypergeometric series from the view point of Weyl group symmetry2006

    • Author(s)
      Masahiko, Ito, Yukari, Sanada
    • Organizer
      the annual meeting of MSJ
    • Place of Presentation
      Osaka Municipal University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] BCn型ジャクソン積分と'基本'対称式2005

    • Author(s)
      伊藤 雅彦
    • Organizer
      RIMS研究集会, 組合せ論的表現論の世界
    • Place of Presentation
      RIMS京都大学
    • Year and Date
      2005-11-11
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] BCn型ジャクソン積分に付随するq-差分deRhamコホモロジーの次元について2005

    • Author(s)
      青本 和彦, 伊藤 雅彦
    • Organizer
      日本数学会秋季分科会
    • Place of Presentation
      岡山大学
    • Year and Date
      2005-09-20
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] vanDiejenのPieri型公式とBCn型ジャクソン積分2005

    • Author(s)
      伊藤 雅彦
    • Organizer
      日本数学会秋季分科会
    • Place of Presentation
      岡山大学
    • Year and Date
      2005-09-20
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] F4型ワイル群不変式の初等的構成法2005

    • Author(s)
      谷口 健二
    • Organizer
      RIMS研究集会, Capelli恒等式の新局面
    • Place of Presentation
      RIMS京都大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] On the rank of the q-twist de Rham cohomology associated with a BCn-type Jackson integral2005

    • Author(s)
      K., Aomoto, M., Ito
    • Organizer
      the annual meeting of MSJ
    • Place of Presentation
      Okayama University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] van Diejen's Pieri formulas and BCn-type Jackson integrals2005

    • Author(s)
      Masahiko, Ito
    • Organizer
      the annual meeting of MSJ
    • Place of Presentation
      Okayama University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] An elementary construction of the invariant polynomials under the action of Weyl group of type F42005

    • Author(s)
      Kenji, Taniguchi
    • Organizer
      RIMS Workshop
    • Place of Presentation
      Kyoto University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] BCn-type Jackson integrals and 'elementary' symmetric polynomials2005

    • Author(s)
      Masahiko, Ito
    • Organizer
      RIMS Workshop
    • Place of Presentation
      Kyoto University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2005-04-01   Modified: 2016-04-21  

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