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Deepening and developing asymptotic representation theory

Research Project

Project/Area Number 16K05164
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionHokkaido University

Principal Investigator

Hora Akihito  北海道大学, 理学研究院, 教授 (10212200)

Project Period (FY) 2016-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords漸近的表現論 / ヤング図形 / 巨大な群 / ランダムウォーク / 自由確率論 / 分岐グラフ / マルコフ連鎖 / 解析学 / 関数解析学 / 確率論 / 表現論
Outline of Final Research Achievements

The fundamental idea of asymptotic representation theory consists of focusing on the statistical feature of representations via the viewpoint of probability theory, which aims at understanding the structure of actions of big groups such as the infinite symmetric group. In this research project, we investigated behavior of statistical ensembles of Young diagrams which often appear in analysis of group representations. We introduced dynamical scaling limit into group-theoretical probabilistic motions in Young diagram ensembles and considered macroscopic time evolution of the limit shapes resulting from the effect of the law of large numbers. We obtained several results on the formulation and the properties for this model.

Academic Significance and Societal Importance of the Research Achievements

ネットワーク科学の急速な発展と相俟って、科学のさまざまな分野においてランダムな性格を有する大規模な系を取り扱う必要が生じているが、このような系の解析は多くの困難を伴う。一方、事物の対称性に着目して複雑な対象にアプローチする数学的手法として、古典的なフーリエ解析から発展した調和解析ないしは群の表現論の方法がある。本研究の大枠である漸近的表現論の基本的なアイデアは、ランダムネスを扱う確率論と対称性に立脚した表現論の融合である。この認識のもと、本研究では、具体的な確率統計モデルに即した研究を通して漸近的表現論の深化と展開を推進し、巨大なランダム系の研究の発展に資することを企図した。

Report

(4 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (13 results)

All 2019 2018 2017 2016

All Presentation (11 results) (of which Int'l Joint Research: 4 results,  Invited: 7 results) Book (2 results)

  • [Presentation] ランダムヤング図形の極限形状の時間発展における微視的待ち時間分布の効果について2019

    • Author(s)
      洞 彰人
    • Organizer
      日本数学会年会
    • Related Report
      2018 Annual Research Report
  • [Presentation] Dynamical scaling limit of the restriction-induction chain on Young diagrams in terms of free probability2018

    • Author(s)
      洞 彰人
    • Organizer
      Random Matrices and Their Applications
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Effect of microscopic pausing time distribution on evolution of macroscopic profiles in Young diagram ensmbles2018

    • Author(s)
      洞 彰人
    • Organizer
      Non-commutative probability and related fields
    • Related Report
      2018 Annual Research Report
  • [Presentation] Dynamical limit shapes for random Young diagrams2018

    • Author(s)
      洞 彰人
    • Organizer
      21th HU-SNU Joint Symposium
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] On evolution of macroscopic profiles and their global fluctuations for growing Young diagrams2018

    • Author(s)
      Akihito Hora
    • Organizer
      第19回北東数学解析研究会
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Markov chains, graph spectra, and some static/dynamic scaling limits2018

    • Author(s)
      洞 彰人
    • Organizer
      第3回代数的組合せ論「仙台勉強会」
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] On a dynamic model for limit profiles and their Gaussian fluctuations in group-theoretical ensembles of Young diagrams2017

    • Author(s)
      洞 彰人
    • Organizer
      Colloquium RIMS Kyoto University
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Dynamic model for limit profiles and their Gaussian fluctuations in Young diagram ensembles2017

    • Author(s)
      Akihito Hora
    • Organizer
      Mathematical Aspects of Quantum Fields and Related Topics
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 群論的なヤング図形集団における巨視的プロファイルとゆらぎの動的モデル2017

    • Author(s)
      洞 彰人
    • Organizer
      筑波大学解析セミナー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] ヤング図形集団における大数の法則(静的および動的モデル)2017

    • Author(s)
      洞 彰人
    • Organizer
      岡山-広島解析・確率論セミナー2017
    • Place of Presentation
      岡山大学理学部(岡山県岡山市)
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Application of free probability to dynamical limit shapes of random Young diagrams2017

    • Author(s)
      Akihito Hora
    • Organizer
      One-day workshop on Interface between Commutative and Non-Commutative Stochastic Analysis
    • Place of Presentation
      北海道大学理学研究院(北海道札幌市)
    • Related Report
      2016 Research-status Report
  • [Book] 対称群の表現とヤング図形集団の解析学2017

    • Author(s)
      洞 彰人
    • Total Pages
      444
    • Publisher
      数学書房
    • ISBN
      9784903342542
    • Related Report
      2017 Research-status Report
  • [Book] The limit shape problem for ensembles of Young diagrams2016

    • Author(s)
      Akihito Hora
    • Total Pages
      73
    • Publisher
      Springer
    • Related Report
      2016 Research-status Report

URL: 

Published: 2016-04-21   Modified: 2020-03-30  

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