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

• Project/Area Number: 21540015
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: University 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
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2009: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: 群の表現論 / 代数的組み合わせ論 / q-series / Macdonald多項式 / Pfaffian / 行列式 / Plane Partitions / Alternating Sign Matrices / 数理物理 / 可解格子モデル / 直行多項式 / 対称群の表現 / Hankel行列式 / パフィアン / 数え上げ組合せ論 / Askey-Willson多項式

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

• [Journal Article] Zeilberger's Holonomic Ansatz for Pfaffians (2012)
  • Author(s): Masao Ishikawa, Christoph Koutschan
  • Journal Title: ISSAC'2012 Volume: (掲載確定)
  • Peer Reviewed
• [Journal Article] A Pfaffian analogue of q-Catalan Hankel determinants and its application (2011)
  • Author(s): Masao Ishikawa and Hiroyuki Tagawa
  • Journal Title: RIMS Kokyuroku Volume: 1738巻 Pages: 102-119
• [Journal Article] A Pfaffian analogue of q-Catalan Hankel determinants and its application (2011)
  • Author(s): Masao Ishikawa, Hiroyuki Tagawa
  • Journal Title: RIMS Kokyuroku Volume: 1738 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
  • NAID: 130004558901
  • Peer Reviewed
• [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 62 Pages: 85-114
  • NAID: 130004558901
  • Peer Reviewed
• [Journal Article] A q-analogue of Catalan Hankel determinant (2009)
  • Author(s): Masao Ishikawa, Hiroyuki Tagawa and Jiang Zeng
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B11 Pages: 11-42
  • Peer Reviewed
• [Journal Article] A q-analogue of Catalan Hankel determinant (2009)
  • Author(s): Msao Ishikawa, Hiroyuki Tagawa, Jiang Zeng
  • Journal Title: RIMS Kokyuroku Bessatsu B11 Pages: 19-42
  • 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 integrals (2011)
  • 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 determinants (2011)
  • 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 the q-Catalan Hankel determinants (2010)
  • 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 Determinant (2010)
  • 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] A Pfaffian Analogue of the q-Catalan Hankel Determinant (2010)
  • 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] (q,t)-hook formula conjecture for d-complete posets and Macdonald polynomials (2010)
  • 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: 2010-08-09
• [Presentation] Fully commutative elements, fully covering elements and d-complete posets (2010)
  • 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] 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] A Pfaffian analogue of q-Catalan Hankel determinants and its application (2010)
  • 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
• [Presentation] A Partial Proof of Okada's(q ; t)-Hook Formula Conjecture for Birds (2010)
  • Author(s): 石川雅雄
  • Organizer: Diagram Algebras and Related Topics
  • Place of Presentation: Culture Resort Festone
• [Presentation] A Partial Proof of Okada's (q ; t)-Hook Formula Conjecture for Birds (2010)
  • Author(s): 石川雅雄
  • Organizer: Diagram Algebras and Related Topics
  • Place of Presentation: Culture Resort Festone
• [Presentation] Enumeration Problem of Certain classes of FPLs and Plane Partitions (2009)
  • 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
• [Presentation] Enumeration Problem of Certain classes of FPLs and Plane Partitions (2009)
  • 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]
• [Remarks]
  • URL: http://www.uec.tottori-u.ac.jp/~mi/
• [Remarks]
  • URL: http://www.uec.tottori-u,ac.jp/~mi/