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Study of asymptotic theory for representations of symmetric groups from the viewpoint of scaling limits for probability models

Research Project

Project/Area Number 16540154
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionOkayama University

Principal Investigator

HORA Akihito  Okayama University, Graduate School of Natural Science and Technology, Associate Professor, 大学院自然科学研究科, 助教授 (10212200)

Co-Investigator(Kenkyū-buntansha) YAMADA Hiro-Fumi  Okayama University, Graduate School of Natural Science and Technology, Professor, 大学院自然科学研究科, 教授 (40192794)
MURAI Joshin  Okayama University, Graduate School of Humanities and Social Sciences, Assistant, 大学院社会文化科学研究科, 助手 (00294447)
SASAKI Toru  Okayama University, Graduate School of Environmental Science, Lecturer, 大学院環境学研究科, 講師 (20260664)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2004: ¥1,300,000 (Direct Cost: ¥1,300,000)
Keywordsprobability model / symmetric group / asymptotic representation theory / quantum probability / random Young diagrams / character of factor representation / Young graph / scaling limit / 量子確率論 / 表現 / 漸近挙動 / 自由確率 / グラフのスペクトル / ユツィス・マーフィー作用素 / 対称群の表現 / ヤング図形 / 自由確率論 / ラプラシアンのスペクトル / 相互作用フォック空間
Research Abstract

The main purpose of the present research is to study asymptotic behavior of various characteristic quantities of representations of symmetric groups and other similar discrete groups as the sizes of the groups grow, and to investigate the limiting pictures from the viewpoint of scaling limits in probability theory and statistical mechanics. Features to be noted in this research include using methods of limit theorems in quantum probability and making much of relations to free probability and random matrices. The following are several concrete results.
1. We studied the spectral distributions of Laplacians with respect to the Gibbs states in zero temperature and infinite volume limit as graphs grow with their degrees and temperatures keeping certain scaling balances. We computed the asymptotic behavior in details under the formulation of quantum central limit theorem by using creation and annihilation operators on interacting Fock spaces.
2. Through combinatorial hard analysis of moments of the Jucys-Murphy element, we studied universal understanding of concentration phenomena in various statistical ensembles consisting of Young diagrams, including those which come from irreducible decomposition of a representation of the symmetric group such as the Littlewood-Richardson coefficients. Many of them are closely related to some properties of random walks on a certain modified Young graph. Here also we applied methods of quantum probability effectively.
3. Under cooperation with T. Hirai and E. Hirai, we constructed a nice factor representation which expresses any character of a wreath product of a compact group with the infinite symmetric group as its matrix element. This representation reflects directly the characterizing parameters for the character beyond a general representation of Gelfand-Raikov.

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (24 results)

All 2006 2005 2004 Other

All Journal Article (24 results)

  • [Journal Article] Realizations of factor representations of finite type with emphasis on their characters for wreath products of compact groups with the infinite symmetric group2006

    • Author(s)
      Hirai, T., Hirai, E., Hora, A.
    • Journal Title

      J. Math. Kyoto Univ. 46-1

      Pages: 75-106

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Jucys-Murphy element and walks on modified Young graph2006

    • Author(s)
      Hora, A.
    • Journal Title

      Banach Center Publ. 73

      Pages: 223-235

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Elementary divisors of Cartan matrices for symmetric groups2006

    • Author(s)
      Uno, K., Yamada, H.-F.
    • Journal Title

      J. Math. Soc. Japan 58-4

      Pages: 1031-1036

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Realizations of factor representations of finite type with emphasis on their characters for wreath products of compact groups with the infinite symmetric group2006

    • Author(s)
      Hirai T., Hirai E., Hora A.
    • Journal Title

      J. Math. Kyoto Univ. 46-1

      Pages: 75-106

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Jucys-Murphy element and walks on modified Young graph2006

    • Author(s)
      Hora A.
    • Journal Title

      Banach Center Publ. 73

      Pages: 223-235

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Elementary divisors of Cartan matrices for symmetric groups2006

    • Author(s)
      Uno K., Yamada H.-F.
    • Journal Title

      J. Math. Soc. Japan 58-4

      Pages: 1031-1036

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Realization of factor representations of finite type with emphasis on their characters for wreath products of compact groups with the infinite symmetric group2006

    • Author(s)
      Hirai, T., Hirai, E., Hora, A.
    • Journal Title

      J. Math Kyoto Univ. 46,no.1

      Pages: 75-106

    • Related Report
      2006 Annual Research Report
  • [Journal Article] An interacting Fock space with periodic Jacobi parameter obtained from regular graphs in large scale limit2006

    • Author(s)
      Akihito Hora, Nobuaki Obata
    • Journal Title

      Quantum Information V

      Pages: 121-144

    • Related Report
      2005 Annual Research Report
  • [Journal Article] An interacting Fock space with periodic Jacobi parameter obtained from regular graphs in large scale limit2005

    • Author(s)
      Hora, A., Obata, N.
    • Journal Title

      Quantum Information 5

      Pages: 121-144

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Remark on Biane's character formula and concentration phenomenon in asymptotic representation theory2005

    • Author(s)
      Hora, A.
    • Journal Title

      Infinite Dimensional Harmonic Analysis 3

      Pages: 141-159

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] 対称群の表現と漸近的組合せ論2005

    • Author(s)
      洞 彰人
    • Journal Title

      数学 57-3

      Pages: 242-254

    • NAID

      10017177056

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Free Probability and Asymptotic Representation Theory of Symmetric Groups2005

    • Author(s)
      洞 彰人
    • Journal Title

      数理解析研究所講究録 1418

      Pages: 10-40

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] An interacting Fock space with periodic Jacobi parameter obtained from regular graphs in large scale limit2005

    • Author(s)
      Hora A., Obata N.
    • Journal Title

      Quantum Information 5

      Pages: 121-144

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Remark on Biane's character formula and concentration phenomenon in asymptotic representation theory2005

    • Author(s)
      Hora A.
    • Journal Title

      Infinite Dimensional Harmonic Analysis 3

      Pages: 141-159

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Remark on Biane's character formula and concentration phenomenon in asymptotic representation theory2005

    • Author(s)
      Akihito Hora
    • Journal Title

      Infinite Dimensional Harmonic Analysis III

      Pages: 141-159

    • Related Report
      2005 Annual Research Report
  • [Journal Article] 対称群の表現と漸近的組合せ論2005

    • Author(s)
      洞 彰人
    • Journal Title

      数学 57/3

      Pages: 242-254

    • NAID

      10017177056

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Free probability and asymptotic representation theory of symmetric groups2005

    • Author(s)
      洞 彰人
    • Journal Title

      数理解析研究所講究録 1418

      Pages: 10-40

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Asymptotic spectral analysis on the Johnson graphs in infinite degree and zero temperature limit2004

    • Author(s)
      Hora, A.
    • Journal Title

      Interdisciplinary Information Sciences 10-1

      Pages: 1-10

    • NAID

      110000071628

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Fat tail phenomena in a stochastic model of stock market : the long-range percolation approach2004

    • Author(s)
      Kuroda, K., Murai, J.
    • Journal Title

      Applied Probab. Trust

    • NAID

      40015872235

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Asymptotic spectral analysis on the Johnson graphs in infinite degree and zero temperature limit2004

    • Author(s)
      Hora A.
    • Journal Title

      Interdisciplinary Information Sciences 10-1

      Pages: 1-10

    • NAID

      110000071628

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Fat tail phenomena in a stochastic model of stock market : the long-range percolation approach2004

    • Author(s)
      Kuroda K., Murai J.
    • Journal Title

      Applied Probab. Trust

    • NAID

      40015872235

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Asymptotic spectral analysis on the Johnson graphs in infinite degree and zero temperature limit2004

    • Author(s)
      Akihito Hora
    • Journal Title

      Interdisciplinary Information Sciences 10

      Pages: 1-10

    • NAID

      110000071628

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Asymptotic spectral analysis of growing regular graphs

    • Author(s)
      Hora, A., Obata, N.
    • Journal Title

      Trans. Amer. Math. Soc. 予定(確約あり)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Asymptotic spectral analysis of growing regular graphs

    • Author(s)
      Hora A., Obata N.
    • Journal Title

      Trans. Amer. Math. Soc, to apper

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary

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Published: 2004-04-01   Modified: 2016-04-21  

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