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Combinatorial Representation Theory which Center of Schur Functions

Research Project

Project/Area Number 15540030
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

YAMADA Hirofumi  OKAYAMA UNIVERSITY, Faculty of Science, Professor, 理学部, 教授 (40192794)

Co-Investigator(Kenkyū-buntansha) YOSHINO Yuji  OKAYAMA UNIVERSITY, Faculty of science, Professor, 理学部, 教授 (00135302)
NAKAMURA Hiroaki  OKAYAMA UNIVERSITY, Faculty of science, Professro, 理学部, 教授 (60217883)
HIRANO Yasuyuki  OKAYAMA UNIVERSITY, Faculty of science, Associate Professor, 理学部, 助教授 (90144732)
TANAKA Katsumi  OKAYAMA UNIVERSITY, Faculty of science, Associate Professor, 理学部, 助教授 (60207082)
IKEDA Takeshi  Okayama University of Science, Faculty of science, Lecturer, 理学部, 講師 (40309539)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2004: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2003: ¥1,500,000 (Direct Cost: ¥1,500,000)
KeywordsSchur functions / Symmetric groups / affine Lie algebras / ヒェルルヒー
Research Abstract

This is an effort for understanding the role of Schur functions and Schur's Q functions, the projective analogue of Schur functions, in representation theory. To be more precise, we proved the following theorem. Schur functions associated with the rectangular Young diagrams occur as weight vectors of the basic representation of the affine Lie algebra of type D^{(2)}_2. And also, they turn out to be the homogeneous tau functions of the nonlinear Schroedinger hierarchy. The key idea for proving the above is to write down the representation spaces and operators in terms of fermions, and derive polynomials via the boson-fermion correspondence. We have succeeded in verifying the similar phenomena for the case of the affine Lie algebra of type A^{(2)}_2. In 2004 we considered the following problem. Clarify the nature of the coefficients in the 2 reduced Schur functions when expanded in terms of Schur's Q functions. Through some experimental computations in small rank cases, I had been convinced that these coefficients are of great interests, both from representation theoretical and combinatorial points of view. Finally we realized that these coefficients are nothing but the so-called Stembridge numbers. As a result we could relate these numbers with the representation theory of affine Lie algebras. Looking carefully at the table of these numbers, we found a simple formula for the elementary divisors of the Cartan matrices of the symmetric groups. More than 10 years ago, Olsson in Copenhagen gave a formula for those, which is expressed in terms of a generating function and is rather complicated. Our version is more direct and combinatorial.

Report

(3 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • Research Products

    (8 results)

All 2004 Other

All Journal Article (6 results) Publications (2 results)

  • [Journal Article] Rectangular Schur functions and fermions2004

    • Author(s)
      池田 岳, 他
    • Journal Title

      Proceedings "FPSAC 04" http://www.pims.math.ca/fpsac/Papers/Ikeda.pdf

      Pages: 8-8

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Rectangular Schur functions and fermions2004

    • Author(s)
      Takeshi IKEDA, Hiroshi MIZUKAWA, Tatsuhiro NAKAJIMA, Hiro-Fumi YAMADA
    • Journal Title

      Proceedings "FPSAC'04"(http://www.pims.math.ca/fpsac/Papers/Ikeda.pdf)

      Pages: 8-8

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Rectangular Schur functions and fermions2004

    • Author(s)
      池田岳 他
    • Journal Title

      Proceedings "FPSAC 04" http://www.pims.math.ca/fpsac/Papers/Ikeda.pdf

      Pages: 8-8

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Rectangular Schur functions and the basic representations of affine Lie algebras

    • Author(s)
      水川裕司, 山田裕史
    • Journal Title

      Discrete Mathematics (to appear)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Rectangular Schur function and the basic representations of affine Lie algebras

    • Author(s)
      Hiroshi MIZUKAWA, Hiro-Fumi YAMADA
    • Journal Title

      Discure mathematics (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Rectangular Schur functions and the basic representations of affine Lie algebras

    • Author(s)
      水川裕司, 山田裕史
    • Journal Title

      Discrete Mathematics (to appear)

    • Related Report
      2004 Annual Research Report
  • [Publications] 水川裕司, 山田裕史: "Rectangular Schur functions and the basic representations of affine hie algebras"Discrete Methematics. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] 池田岳 他: "Rectangular Schur functions and fermions"Proceadings "International Conference on Formal Power Series and Algebraic combinatorics". (to appear).

    • Related Report
      2003 Annual Research Report

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

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