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2011 Fiscal Year Final Research Report

The determinants and Pfaffians appearing in the enumeration of plane partitions and its applications on mathematical physics

Research Project

  • PDF
Project/Area Number 21540015
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionUniversity of the Ryukyus (2011)
Tottori University (2009-2010)

Principal Investigator

ISHIKAWA Masao  琉球大学, 教育学部, 教授 (40243373)

Co-Investigator(Renkei-kenkyūsha) OKADA Soichi  名古屋大学, 大学院・多元数理科学研究科, 教授 (20224016)
Project Period (FY) 2009 – 2011
Keywords群の表現論 / 代数的組み合わせ論
Research Abstract

In the enumeration problems of plane partitions under several symmetries and their generalizations we obtain several interesting determinants and Pfaffians as their generating functions. We have been interested in the evaluation of these determinants and Pfaffians and have been intensively studied those evaluation problems. Several studies have been hinted close relations between the evaluation of Hankel determinants and so-called Hankel Pfaffians and the evaluation of the determinants and Pfaffians appearing in the enumeration problems. In our recent study we find a method to evaluate Hankel Pfaffians by the Selberg integrals through de Bruijn's formula. In fact it is well-known that the Selberg integrals and its extensions are closely related to the orthogonal polynomials and Dunkl operators and its evaluation can be obtained by multivariate generalizations of the classical orthogonal polynomials. Further we can relate the Hankel determinants and Pfaffians with the determinants and Pfaffians appearing in our enumeration problems by manipulations of the generating functions of the entries of determinants(Pfaffians). Especially the associated Jacobi polynomials play an important role in the enumeration of the plane partitions because its distribution measure is related to Gauss's hypergeometric series_3F2. We consider Hankel determinants or Pfaffians whose entries are obtained from moments of a class of orthogonal polynomials. The moment generating function related to our enumeration problems seem to be related to the associated Jacobi polynomials by considering the continued fraction expansion of the moment generating function related to our enumeration problems.

  • Research Products

    (14 results)

All 2011 2010 2009 Other

All Journal Article (4 results) (of which Peer Reviewed: 3 results) Presentation (9 results) Remarks (1 results)

  • [Journal Article] A Pfaffian analogue of q-Catalan Hankel determinants and its application2011

    • Author(s)
      Masao Ishikawa and Hiroyuki Tagawa
    • Journal Title

      RIMS Kokyuroku

      Volume: 1738巻 Pages: 102-119

  • [Journal Article] 行列式・パフィアンに関する等式とその表現論,組合せ論への応用2010

    • Author(s)
      石川雅雄・岡田聡一
    • Journal Title

      数学

      Volume: 62巻 Pages: 85-114

    • Peer Reviewed
  • [Journal Article] A q-analogue of Catalan Hankel determinant2009

    • Author(s)
      Masao Ishikawa, Hiroyuki Tagawa and Jiang Zeng
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B11 Pages: 11-42

    • Peer Reviewed
  • [Journal Article] Zeilberger's Holonomic Ansatz for Pfaffians

    • Author(s)
      Masao Ishikawa and Hiroyuki Tagawa
    • Journal Title

      ISSAC' 2012

      Volume: (掲載予定)

    • Peer Reviewed
  • [Presentation] A Pfaffian analogue of the Hankel determinants and the Selberg integrals2011

    • Author(s)
      石川雅雄
    • Organizer
      Topics in Combinatorial Representation Theory
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2011-10-14
  • [Presentation] Pfaffian decomposition and a Pfaffian analogue of q-Catalan Hankel determinants2011

    • Author(s)
      Jiang Zeng
    • Organizer
      International Conference on Asymptotics and Special Functions
    • Place of Presentation
      香港市立大学
    • Year and Date
      2011-05-30
  • [Presentation] A Pfaffian analogue of q-Catalan Hankel determinants and its application2010

    • Author(s)
      石川雅雄
    • Organizer
      組合せ論的表現論とその応用
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      20101019-22
  • [Presentation] A Partial Proof of Okada's(q ; t)-Hook Formula Conjecture for Birds2010

    • Author(s)
      石川雅雄
    • Organizer
      Diagram Algebras and Related Topics
    • Place of Presentation
      Culture Resort Festone
    • Year and Date
      20100705-09
  • [Presentation] A Pfaffian analogue of the q-Catalan Hankel determinants2010

    • Author(s)
      石川雅雄
    • Organizer
      Combinatoire et Theorie des Nombres
    • Place of Presentation
      Institut Camille Jordan, Universite Claude Bernard Lyon 1
    • Year and Date
      2010-09-28
  • [Presentation] A Pfaffian Analogue of the q-Catalan Hankel Determinant2010

    • Author(s)
      石川雅雄
    • Organizer
      65th S' eminaire Lotharingien de Combinatoire
    • Place of Presentation
      Strobl(Wolfgangsee), Austria
    • Year and Date
      2010-09-13
  • [Presentation] Fully commutative elements, fully covering elements and d-complete posets2010

    • Author(s)
      石川雅雄
    • Organizer
      Seminar on analysis of combinatorial structures and its applications for information theory
    • Place of Presentation
      ニューミヤコホテル足利
    • Year and Date
      2010-08-07
  • [Presentation] A Partial Proof of Okada's(q ; t)-Hook Formula Conjecture for Birds(and Insets)2010

    • Author(s)
      石川雅雄
    • Organizer
      64th Seminaire Lotharingien de Combinatoire
    • Place of Presentation
      Universite Claude Bernard Lyon
    • Year and Date
      2010-03-30
  • [Presentation] Enumeration Problem of Certain classes of FPLs and Plane Partitions2009

    • Author(s)
      石川雅雄
    • Organizer
      Combinatorics Summer School 2009(COS 2009)
    • Place of Presentation
      稚内北星学園大学
    • Year and Date
      2009-09-02
  • [Remarks]

    • URL

      http://www.uec.tottori-u.ac.jp/~mi/

URL: 

Published: 2013-07-31  

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