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Study on representation theory of the classical groups and their related combinatorics

Research Project

Project/Area Number 18540046
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

KOIKE Kazuhiko  Aoyama Gakuin University, College of Science and Engineering, Professor (70146306)

Co-Investigator(Kenkyū-buntansha) ITO Masahiko  Aoyama Gakuin University, College of Science and Engineering, Associate Professor (30348461)
TANIGUCHI Kenji  Aoyama Gakuin University, College of Science and Engineering, Associate Professor (20306492)
Project Period (FY) 2006 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥4,010,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥510,000)
Fiscal Year 2007: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2006: ¥1,800,000 (Direct Cost: ¥1,800,000)
Keywordsalgebra / combinatoics / representation theor / classical groups / Weyl groups / 組み合せ論 / 対称式 / ヘッケ環
Research Abstract

For the classical Lie groups, the famous formulas by Herman Weyl, which say that the denominators of their irreducible characters are completely decomposed into the product of the linear factors, are regarded as the determinants whose components are the irreducible characters of rank one Lie groups of the same type. From this view point, Koike and Ito generalize these formulas and give some formulas that the determinants, whose components (the row indices are parameterized by the Young diagrams included in some fixed rectangular and the column indices are parameterized by the subsets of the variables) are given by the irreducible characters corresponding to the Young diagrams of the row indices in their variables prescribed by the column indices, are completely decomposed into the products of the linear factors. The resulting formulas turn out to be the powers of the original denominator formulas. These formulas play an crucial role in the result by Aomoto-Ito, in which they construct the fundamental system of solutions of holonmic q-difference system of Jackson integrals of type BC_n. Ito also gives explicit connection formulas of the solutions of holonmic q-difference system of Jackson integrals of type BC_n. Taniguchi gives an elementary construction for the invariant polynomials of type F_4 under the action of its Weyl group, using the degree 2 invariant polynomials under the action of Weyl group of type D_4, based on the fact that Weyl group of type D(-4) is included in a normal subgroup of Weyl group of type F_4

Report

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

    (32 results)

All 2008 2007 2006 Other

All Journal Article (23 results) (of which Peer Reviewed: 10 results) Presentation (9 results)

  • [Journal Article] A multiple generalization of Slater's transformation formula for a very-well-poised-balanced _<2r>φ_<2r> series2008

    • Author(s)
      伊藤 雅彦
    • Journal Title

      Quarterly Journal of Math (in press)

    • 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 Sears-Slater basic hypergeometric transformations2008

    • Author(s)
      伊藤 雅彦, 真田 ゆかり
    • Journal Title

      Ramanujan J. (in press)

    • 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 generalizedfrom Gustafson's Cn-type sum2008

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

      Journal of Difference Equations and Applications (accepted)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A multiple generalization of Slater's transformation formula for a very-well-poised-balanced 2rφ2r series2008

    • Author(s)
      Masahiko Ito
    • Journal Title

      The Quarterly Journal of Mathematics (in press)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A generalization of Weyl's denominator formulas for the classical groups2006

    • Author(s)
      小池 和彦、伊藤 雅彦
    • 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)
      伊藤 雅彦
    • 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'elementar y'symmetric polynomials2006

    • Author(s)
      伊藤 雅彦
    • 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'element ary'symmetric polynomials2006

    • Author(s)
      伊藤 雅彦
    • 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] BC_n型ジャクソン積分ど'基本'対称式2006

    • Author(s)
      伊藤 雅彦
    • Journal Title

      京都大学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
    • Peer Reviewed
  • [Journal Article] F_4型ワイル群不変式の初等的構成法2006

    • Author(s)
      谷口 健二
    • Journal Title

      京都大学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
    • Peer Reviewed
  • [Journal Article] A generalization of Weyl's denominator formulas for the classical groups2006

    • Author(s)
      Kazuhiko, Koike, Masahikom, 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 C_n-type summation formula via 'elementary' symmetric polynomials2006

    • Author(s)
      Masahiko, Ito
    • Journal Title

      Publications of RIMS 42

      Pages: 523-549

    • NAID

      110004626193

    • 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 BC_n type Jackson integral from 'elementary' symmetric polynomials2006

    • Author(s)
      Masahiko, Ito
    • Journal Title

      Adv. in Math 204

      Pages: 619-646

    • 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 BC_n 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] an elementary construction of the invariant polynomials under the action of Weyl group of type F_42006

    • 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] A generalization of Weyl's denominator formulas for the classical groups2006

    • Author(s)
      小池和彦, 伊藤雅彦
    • Journal Title

      Journal of Algebra 302

      Pages: 817-825

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

    • Author(s)
      伊藤雅彦
    • Journal Title

      Ramanujan Journal 12

      Pages: 131-151

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

    • Author(s)
      伊藤雅彦
    • Journal Title

      Publications of RIMS 42

      Pages: 523-549

    • Related Report
      2006 Annual Research Report
  • [Journal Article] q-difference shift for a BCn type Jackson integral arising from 'elementary' symmetric polynomials2006

    • Author(s)
      伊藤雅彦
    • Journal Title

      Advances in Mathematics 204

      Pages: 619-646

    • 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

      数理研講究録 1497

      Pages: 79-87

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

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

      Ramanujan J (in press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related 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
      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 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] 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
    • Related Report
      2007 Annual Research Report
  • [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
    • 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
    • Place of Presentation
      Saitama University
    • 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] G_2型マクドナルド対称多項式のピェリ公式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 G_22006

    • 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 formule 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

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Published: 2006-04-01   Modified: 2016-04-21  

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