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Combinatorial Representation Theory of Affine Lie Alqebras and Symmetric Groups

Research Project

Project/Area Number 11640001
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionOKAYAMA UNIVERSITY (2000)
Hokkaido University (1999)

Principal Investigator

YAMADA Hiro-fumi  Faculty of Science, OKAYAMA UNIVERSITY, Professor, 理学部, 教授 (40192794)

Co-Investigator(Kenkyū-buntansha) 田口 雄一郎  北海道大学, 大学院・理学研究科, 助教授 (90231399)
斎藤 睦  北海道大学, 大学院・理学研究科, 助教授 (70215565)
山下 博  北海道大学, 大学院・理学研究科, 助教授 (30192793)
中島 達洋  明海大学, 経済学部, 講師 (00286006)
渋川 陽一  北海道大学, 大学院・理学研究科, 助手 (90241299)
Project Period (FY) 1999 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2000: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 1999: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsAffine Lie Alqebras / Symmetric Groups / Schur Functions / Young Diagrams / アフィンリー環 / 指標
Research Abstract

My first attempt was to describe the weight basis of the basic representations of several typical affine Lie algebras. In particular, for the simplest affine Lie algebra A^<(1)>_1, I considered two realizations of the basic representation and found that the modular version of the Schur functions and Schur's Q-functions occur as weight basis, respectively. Analysing these two realizations, I found an interesting phenomenon for the elementary divisors of the spin decomposition matrices for the symmetric group. Namely the elemntary divisors of the spin decomposition matrices for prime 2 are all powers of 2. Though this fact actually can be proved by a general theory of modular representations, I could give a simple proof of this by utilizing representations of the affine Lie algebra A^<(1)>_1.
Studying the zonal polynomials, which are a specialization of the Jack polynomials, I found an interesting fact in the character tables of the symmetric group. Later I recognizes that this fact had been found more than 50 years ago by Littlewood, whose proof is a bit complicated. I gave a simple proof of this fact as well as its spin version with Hiroshi Mizukawa, a graduate student. The main tools for the proof are again Schur functions and Schur's Q-functions.
In the joint work with Takeshi Ikeda I could obtain all the homogeneous polynomial solutions for the nonlinear Schrodinger hierarchy. The schur functions indexed by the rectangular Young diagrams play an essential role in this theory.

Report

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

    (9 results)

All Other

All Publications (9 results)

  • [Publications] 山田裕史: "Schur functions and two realizations of the basic A^<(1)>_1-module"Proc.Int'l Workshop on Special Functions. 431-438 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 中島達洋,山田裕史: "Schur's Q-functions and twisted affine Lie algebras"Advanced Studies in Pure Mathematics. 28. 241-259 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Hiro-Fumi Yamada: "Schur function and two realizations of the basic A^<(1)>_1-modules"Proceedings of the International Workshop on Special Functios. 431-438 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Tatsuhiro Nakajuna and Hiro-Fumi Yamada: "Sclur's Q-functions and twisted affine Lie algebras"Advanced Studies in Pare Mathematics. 28. 241-259 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 山田裕史: "Schur functions and two realizations of the basic A_l^<(1)>-module"Proc.Int'l Workshop on Special Functions. 431-438 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 中島達洋,山田裕史: "Schur's Q-functions and twisted affine Lie algebras"Adv.Studies in Pure Math.. 28. 241-259 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] YAMADA, Hiro-Fumi: "Reduced Schur Functions and Littlewood-Richardson coeffcients"Journal of London Mathematical Society. (2)59. 396-406 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] YAMADA, Hiro-Fumi: "Schur's Q-functions and twisted affine Lie algelras"Advanced Studies in Pure Mathematics. (印刷中).

    • Related Report
      1999 Annual Research Report
  • [Publications] YAMADA, Hiro-Fumi: "Schur functions and two realizations of the basic A^1_<(1)>-module"Proceedings of the Conf. on "Special Functions". (印刷中).

    • Related Report
      1999 Annual Research Report

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

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