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A Combinatorial Representaion approach to Iwahori Whittaker function

Research Project

Project/Area Number 15K17519
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Algebra
Research InstitutionSophia University

Principal Investigator

NAKASUJI Maki  上智大学, 理工学部, 准教授 (30609871)

Research Collaborator BUMP Daniel  
NARUSE Hiroshi  
Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
KeywordsWhittaker関数 / Hecke環 / Casselman基底 / Casselman問題 / Kazhdan-Lusztig多項式 / Schur多重ゼータ関数 / 行列式表示 / Iwahori Hecke環 / Iwahori Whittaker関数 / Yang-Baxter基底
Outline of Final Research Achievements

(1) Casselman basis is a basis vectors of the Iwahori Whittaker functions. Casselman problem is to get a closed formula for the change of basis between the Casselman basis and the natural basis. We obtained the solution to this problem by using the Yang-Baxter basis of Hecke algebra (joint work with H. Naruse).
(2) We introduced the deformation of the Kazhdan-Lusztig R-polynomial. As an application of it, under some assumption we proved Bump-Nakasuji conjecture which is the closed formula for the transition matrices between the deformation of the Casselman basis and natural basis, and obtained certain new functional equations for this transition matrices (joint work with D. Bump).

Academic Significance and Societal Importance of the Research Achievements

Whittaker関数と組合せ論的表現論の連携は,2013 年頃から始まった新しい取り組みである.それぞれの理論には,Demazure 作用素や Bruhat 順序などの共通の道具があることは確認しているが,実際に領域間の関係を持ち込む研究は少ない.これに対し本研究は,Whittaker 関数の問題に対して,シューベルトカルキュラスやKazhdan-Lusztig多項式といった組合せ論的表現論を持込み問題の解決を試みた. 本研究は理論間の関係構築に貢献し,連携をより確実なものとしたと言える. また,これにより今後の解析数論およびその周辺の性質の解明に対する新しい展望を開くことが期待できる.

Report

(4 results)
  • 2018 Final Research Report ( PDF )
  • 2017 Annual Research Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (30 results)

All 2018 2017 2016 2015 Other

All Int'l Joint Research (3 results) Journal Article (10 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 4 results,  Acknowledgement Compliant: 6 results) Presentation (17 results) (of which Int'l Joint Research: 3 results,  Invited: 1 results)

  • [Int'l Joint Research] Stanford大学(米国)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] Chulalongkorn University(タイ)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] Chulalongkorn University(タイ)

    • Related Report
      2015 Research-status Report
  • [Journal Article] On Schur multiple zeta functions: A combinatoric generalization of multiple zeta functions2018

    • Author(s)
      Maki Nakasuji, Ouamporn Phuksuwan and Yoshinori Yamasaki
    • Journal Title

      Advances in Mathematics

      Volume: 印刷中

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Casselman’s Basis of Iwahori vectors and Kazhdan-Lusztig polynomials2018

    • Author(s)
      Daniel Bump and Maki Nakasuji
    • Journal Title

      Canadian Journal of Mathematics

      Volume: 印刷中

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Skew Schur 多重ゼータ関数とその行列式表示2017

    • Author(s)
      中筋麻貴
    • Journal Title

      「第十回数論女性の集まり」報告集

      Volume: 1 Pages: 61-69

    • Related Report
      2017 Annual Research Report
  • [Journal Article] 局所体上の主系列表現の基底2017

    • Author(s)
      中筋麻貴
    • Journal Title

      第14回城崎新人セミナー報告集

      Volume: -

    • Related Report
      2017 Annual Research Report 2016 Research-status Report
  • [Journal Article] Yang-Baxter basis of Hecke algebra and Casselman's problem (extended abstract)2016

    • Author(s)
      Maki Nakasuji and Hiroshi Naruse
    • Journal Title

      Discrete Mathematics and Theoretical Compute Science

      Volume: 1 Pages: 935-946

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Casselman基底と自然基底の変換係数2016

    • Author(s)
      中筋麻貴
    • Journal Title

      「第九回数論女性の集まり」報告集

      Volume: 1 Pages: 90-96

    • Related Report
      2016 Research-status Report
    • Acknowledgement Compliant
  • [Journal Article] Jacobi-Trudi formula for some feature of Schur multiple zeta functions2016

    • Author(s)
      Ouamporn Phuksuwan and Maki Nakasuji
    • Journal Title

      「第九回数論女性の集まり」報告集

      Volume: 1 Pages: 117-122

    • Related Report
      2016 Research-status Report
    • Acknowledgement Compliant
  • [Journal Article] Yang-Baxter basis of Hecke algebra and Casselman's problem (extended abstract)2016

    • Author(s)
      Maki Nakasuji, Hiroshi Naruse
    • Journal Title

      Discrete Mathematics and Theoretical Computer Science

      Volume: 1

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Casselman問題へのアプローチ法2015

    • Author(s)
      中筋麻貴
    • Journal Title

      「第八回数論女性の集まり」報告集

      Volume: 1 Pages: 63-71

    • Related Report
      2015 Research-status Report
    • Acknowledgement Compliant
  • [Journal Article] K-theoretic Schur P-function and set-valued decomposition tableaux2015

    • Author(s)
      Maki Nakasuji, Takeshi Ikeda
    • Journal Title

      第17回代数群と量子群の表現論研究集会報告集

      Volume: 1 Pages: 106-115

    • Related Report
      2015 Research-status Report
    • Acknowledgement Compliant
  • [Presentation] Casselman基底に関する変換係数について2018

    • Author(s)
      中筋麻貴,Daniel Bump
    • Organizer
      日本数学会
    • Related Report
      2017 Annual Research Report
  • [Presentation] Schur multiple zeta functions2017

    • Author(s)
      Maki Nakasuji
    • Organizer
      French-Japanese Zeta Functions
    • Place of Presentation
      Lille 1 University (France)
    • Year and Date
      2017-03-13
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] 局所体上の主系列表現の基底-Hecke環とIwahori Whittaker関数からのアプローチ-2017

    • Author(s)
      中筋麻貴
    • Organizer
      第14回城崎新人セミナー
    • Place of Presentation
      城崎市民センター大会議室(兵庫県・豊岡市)
    • Year and Date
      2017-02-16
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Schur多重ゼータ関数の行列式表示2017

    • Author(s)
      中筋麻貴
    • Organizer
      第35回関西多重ゼータ研究会
    • Related Report
      2017 Annual Research Report
  • [Presentation] Skew Schur 多重ゼータ関数とその行列表示2017

    • Author(s)
      中筋麻貴
    • Organizer
      第10回数論女性の集まり
    • Related Report
      2017 Annual Research Report
  • [Presentation] Schur type multiple zeta functions and its determinant formulae2017

    • Author(s)
      Maki Nakasuji
    • Organizer
      Algebraic Lie Theory and Representation Theory 2017
    • Related Report
      2017 Annual Research Report
  • [Presentation] Schur多重ゼータ関数とSchur多重ゼータ値の山本積分表示2017

    • Author(s)
      中筋麻貴
    • Organizer
      早稲田大学整数論セミナー
    • Related Report
      2017 Annual Research Report
  • [Presentation] Schur multiple zeta functions2017

    • Author(s)
      Maki Nakasuji
    • Organizer
      Number Theory and Combinatorics Seminar, Stanford
    • Related Report
      2017 Annual Research Report
  • [Presentation] Schur型多重ゼータ関数に対するJacobi-Trudi公式2016

    • Author(s)
      中筋麻貴
    • Organizer
      無限可積分セミナー
    • Place of Presentation
      神戸大学(兵庫県・神戸市)
    • Year and Date
      2016-11-16
    • Related Report
      2016 Research-status Report
  • [Presentation] Schur多重ゼータ値および擬対称関数のJacobi-Trudi型公式について2016

    • Author(s)
      中筋麻貴
    • Organizer
      解析数論セミナー
    • Place of Presentation
      名古屋大学(愛知県・名古屋市)
    • Year and Date
      2016-10-14
    • Related Report
      2016 Research-status Report
  • [Presentation] Schur多重ゼータ関数のJacobi-Trudi型公式2016

    • Author(s)
      中筋麻貴
    • Organizer
      日本数学会年会
    • Place of Presentation
      関西大学(大阪府・吹田市)
    • Year and Date
      2016-09-17
    • Related Report
      2016 Research-status Report
  • [Presentation] Casselman's basis, Yang-Baxter basis, and Kostant Kumar's twisted group algebra2016

    • Author(s)
      Maki Nakasuji
    • Organizer
      Whittaker Functions: Number Theory, Geometry , and Physics
    • Place of Presentation
      Banff International Research Station (Canada)
    • Year and Date
      2016-07-28
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Casselman基底と自然基底の変換係数2016

    • Author(s)
      中筋麻貴
    • Organizer
      第9回数論女性の集まり
    • Place of Presentation
      上智大学(東京都・千代田区)
    • Year and Date
      2016-05-21
    • Related Report
      2016 Research-status Report
  • [Presentation] Casselman問題とIwahori Hecke環2015

    • Author(s)
      中筋麻貴
    • Organizer
      西早稲田数論セミナー
    • Place of Presentation
      早稲田大学(東京都・ 新宿区)
    • Year and Date
      2015-12-19
    • Related Report
      2015 Research-status Report
  • [Presentation] Casselman問題とDuality2015

    • Author(s)
      中筋麻貴,成瀬弘
    • Organizer
      日本数学会2015年度秋季総合分科会
    • Place of Presentation
      京都産業大学(京都府・ 京都市)
    • Year and Date
      2015-09-15
    • Related Report
      2015 Research-status Report
  • [Presentation] Casselman's basis of Iwahori fixed vectors2015

    • Author(s)
      Maki Nakasuji
    • Organizer
      French-Japanese Workshop on multiple zeta functions and applications
    • Place of Presentation
      University Jean Monnet (フランス・ Saint-Etienne)
    • Year and Date
      2015-09-09
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] Casselman問題へのアプローチ法2015

    • Author(s)
      中筋麻貴
    • Organizer
      研究集会「第8回数論女性の集まり」
    • Place of Presentation
      上智大学(東京都・ 千代田区)
    • Year and Date
      2015-05-30
    • Related Report
      2015 Research-status Report

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Published: 2015-04-16   Modified: 2022-06-07  

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