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Representation theory and combinatorics of classical groups, quantum groups and Hecke algebras

Research Project

Project/Area Number 12640011
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionThe University of Tokyo

Principal Investigator

TERADA Itaru  University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (70180081)

Co-Investigator(Kenkyū-buntansha) OKADA Soichi  Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (20224016)
TANAKA Yohei  Tokyo University of Mercantile Marine, Department of Mathematics, Associate Professor, 商船学部, 助教授 (00135295)
KOIKE Kazuhiko  Aoyama Gakuin University, Department of Mathematics, Professor, 理工学部, 教授 (70146306)
OCHIAI Hiroyuki  Tokyo Institute of Technology, Department of Mathematics, Associate Professor, 大学院・理工学研究科, 助教授 (90214163)
KOBAYASHI Toshiyuki  University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (80201490)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2001: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2000: ¥1,800,000 (Direct Cost: ¥1,800,000)
Keywordscombinatorics / representation theory / Young diagrams / the Robinson-Schensted correspondence / the Springer variety / partially ordered sets / involutions / Brauer diagrams / Springer varieyy / Brauer diagram / ロビンソン・シェンステッド対応 / tableau / Steinberg variety / ジョルダン標準形
Research Abstract

Throughout the two years, we investigated the possibility of providing some combinatorial interpretation to the "generalized Robinson-Schensted correspondence for real Lie groups" in the case where the group is SU^*(2n), explicitly described by P. Trapa. This is a bijection between the so-called Brauer diagrams on 2n points (which can also be interpreted as the fixed-point-free involutions) and the standard Young tableaux with 2n boxes whose column lengths are all even. One possible approach is, utilizing the fact that the classical Robinson-Schensted correspondence gives a bijection between the same sets, to find a transformation on Brauer diagrams which, when composed with the classical Robinson-Schensted correspondence, gives Trapa's bijection. Another possible approach is to characterize Trapa's bijection using some combinatorial quantities, in the same sense that the classical Robinson-Schensted correspondence can be characterized by certain invariants of posets due to Greene and … More Kleitman. In the first year, we obtained a conjecture in the second direction. We are still in pursuit of this conjecture Through interaction with various researchers, both domestic and overseas, we have received several interesting suggestions. One is to look for further possibility of extending our geometric interpretation to the Knuth version of the updown Robinson-Schensted correspondence devised by Pak and Postnikov. There has been another suggestion, somewhat related, that the irreducible decomposition of the variety of N-stable flags with gaps in dimensions is not yet clear. In view of these suggestions, it looks also necessary to find new approaches to combinatorial interpretations through generalization to the Knuth version, possibly as well as piecewise linear version. On the other hand, we continued to support T. Roby's research on constructing bijections giving the formal character identities corresponding to the decomposition of the tensor powers of the Weil representation of Sp(2n, R). This has come to a conclusion in the case n = 2. Less

Report

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

    (27 results)

All Other

All Publications (27 results)

  • [Publications] 寺田 至: "Brauer diagrams, updown Tableaux and nilpotent matrices"J. Algebraic Combin.. 14. 229-267 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 小池 和彦: "Spin representations and centralizer algebras far Spin(2n+1)"京都大学数理解析研究所講究録. (掲載予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 小池 和彦: "Spin representations and centralizer algebras far Spin(2n)"京都大学数理解析研究所講究録. (掲載予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 小林 俊行: "Discretely decomposable restrictions of unitary representations of reductive Lie groups-examples and conjectures"Analysis on homogeneous space and representation theory (Adv. Stud. Pure Math.). 26. 98-126 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 落合 啓之: "On the power of a prime p deviding the numper of solutions of x^<p=1> in a symmetric group"Annals of Combin.. 5. 197-210 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 山口 学: "A duality of a twisted group algebra of the hyperoctahedral group and the queer Lie superalgebra"Combinatorial methods in Representation theory (Adu. Stng. Pure Math). 28. 401-422 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 柏原正樹, 小池和彦, 岡田聡一, 寺田 至, 山田裕文(編): "Combinatorial methods in Representation theory"紀伊国屋書店. 422 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Itaru Terada: "Brauer diagrams, updown tableaux, and nilpotent matrices"J. Algebraic Combin. 14. 229-267 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kazuhiko Koike: "Spin representations and centralizer algebras for Spin(2n + 1)"Surikaisekikenkyusho Kokyuroku (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Kazuhiko Koike: "Spin representations and centralizer algebras for Spin(2n)"Surikaisekikenkyusho Kokyuroku (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Toshiyuki Kobayashi: "Discretely decomposable restrictions of unitary representations of reductive Lie groups - examples and conjectures"Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Okayama-Kyoto (Adv. Stud. Pure Math., vol.26). 98-126 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroyuki Ochiai (with H. Ishihara, Y. Takegahara, T. Yoshida): "On the power of a prime p dividing the number of solutions of x^p = 1 in a symmetric group"Annals of Comb. 5. 197-210 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Manabu Yamaguchi: "A duality of a twisted group algebra of the hyperoctahedral group and the queer Lie superalgebra"Combinatorial Methods in Representation Theory, Kyoto (Adv. Stud. Pure Math., vol.28). 401-422 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Masaki Kashiwara, Kazuhiko Koike, Soichi Okada, Itaru Terada, Hiro-Fumi Yamada (eds.): "Combinatorial Methods in Representation Theory, Kyoto (Adv. Stud. Pure Math., vol.28)"Kinokuniya, Tokyo. (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Toshiyuki Kobayashi, et al. (eds.): "Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Okayama-Kyoto (Adv. Stud. Pure Math., vol.26)"Kinokuniya, Tokyo. (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 寺田 至: "Brauer diagrams, updown tableaux and nilpotent matrices"J. Algebraic Combin. 14・3. 229-267 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 小池和彦: "Spin representations and contralizer algebras for Spin(2n+1)"京都大学数理解析研究所講究録. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] 小池和彦: "Spin representations and contralizer algebras for Spin(2n)"京都大学数理解析研究所講究録. (発表予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] 小林俊行: "Discretely decomposable restrictions of unitary representations of reductive Lie groups-examples and conjectures"論文集 Analysison homogeneous spaces and representation theory(Advanced studies in Pure Mathematics). 26. 98-126 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] 落合啓之: "On the Power of a primep deviding the number of solutions of XP=1 in a symmetric group"Annals of Combin. 5. 197-210 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 山口 学: "A duality of a twisted group algebra of the hyperoctahedral group and the gueer Lie superalgebra"論文集Combinatorial methods in representation theory(Advanced studies in Pure Mathematics). 28. 401-422 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] 小林俊行: "Discretely decomposable restrictions of unitary representations of reductive Liegro-pc-examples and carjectures"Advanced Studies in Pura Mathematics (論文集シリーズ). 26. 98-126 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 小林俊行: "Discontinuous groups for non-Riemannian homogeneous spaces"Mathematics Unlimited-2001 and beyond . (論文集,Springer-Verlag発行). 723-748 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 落合啓之: "Classification of commuting differential operators with two varlables."Yang-Baxter systems, non-linear models and their applications . (論文集,World Scientific発行). 183-187 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 落合啓之: "The degree of orbits of multiplicity-free actions"Asterisque. 279(発表予定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] 落合啓之: "Non-commutative harmonic escillators and Fuchsian ordinary differential operators"Communications in Mathematical Physics. (発表予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] 山口学: "A dualrty of a twisted group algebra of the Weyl group of type Bn and the queer Lie superalgebra"Combinatorial methods in representation theory. (論文集,紀伊國屋発行). 401-422 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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