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

平面分割枚举中出现的行列式和普法夫式及其在数学物理中的应用

• 批准号: 21540015
• 负责人: ISHIKAWA Masao
• 金额: $ 2.91万
• 依托单位: University of the Ryukyus (2011)Tottori University (2009-2010)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2009
• 资助国家: 日本
• 起止时间: 2009 至 2011
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-21540015/
• 关键词: 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

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.

在几种对称性及其推广下的平面划分计数问题中，我们得到了几个有趣的行列式和Pfaffian作为它们的母函数。我们一直对这些行列式和Pfaffian的评价感兴趣，并对这些评价问题进行了深入的研究。一些研究表明，汉克尔行列式和所谓的汉克尔Pfaffian的求值与计数问题中出现的行列式和Pfaffian的求值之间存在密切的关系。在我们最近的研究中，我们通过de Bruijn公式找到了一种通过Selberg积分来估计Hankel Pfaffian的方法。实际上，Selberg积分及其推广与正交多项式和Dunkl算子密切相关，它的计算可以通过对经典的正交多项式的多元推广而得到。此外，我们可以通过操纵行列式(Pfaffian)项的母函数，将Hankel行列式和Pfaffian与我们的计数问题中出现的行列式和Pfaffian联系起来。特别是伴随的Jacobi多项式在平面划分的计数中起着重要的作用，因为它的分布测度与Gauss的超几何级数3F2有关。我们考虑Hankel行列式或Pfaffian行列式，它们的项由一类正交多项式的矩得到。通过考虑与计数问题有关的矩母函数的连分式展开式，与计数问题有关的矩母函数似乎与相关的雅可比多项式有关。

项目成果

A Pfaffian analogue of the q-Catalan Hankel determinants

q-Catalan Hankel 行列式的普法夫类似物

• 发表时间: 2010
• 作者: Msao Ishikawa;Hiroyuki Tagawa;Jiang Zeng;Hiroyuki Yoshida;谷川好男,古屋淳;Masao Ishikawa and Hiroyuki Tagawa;Hiroyuki Yoshida;Y. Tanigawa;石川雅雄;Y.Tanigawa;Hiroyuki Yoshida;Jiang Zeng;Y. Tanigawa;Hiroyuki Yoshida;石川雅雄
• 通讯作者: 石川雅雄

A Partial Proof of Okada's (q ; t)-Hook Formula Conjecture for Birds (and Insets)

冈田 (q ; t)-Hook 鸟类公式猜想的部分证明（和插图）

• 发表时间: 2010
• 作者: 青影一也;水川裕司;山田裕史;石川雅雄
• 通讯作者: 石川雅雄

A Pfaffian analogue of the Hankel determinants and the Selberg integrals

汉克尔行列式和塞尔伯格积分的普法夫模拟

• 发表时间: 2011
• 作者: Msao Ishikawa;Hiroyuki Tagawa;Jiang Zeng;Hiroyuki Yoshida;谷川好男,古屋淳;Masao Ishikawa and Hiroyuki Tagawa;Hiroyuki Yoshida;Y. Tanigawa;石川雅雄
• 通讯作者: 石川雅雄

Fully commutative elements, fully covering elements and d-complete posets

完全交换元素、完全覆盖元素和 d-完全偏序集

• 发表时间: 2010
• 作者: Msao Ishikawa;Hiroyuki Tagawa;Jiang Zeng;Hiroyuki Yoshida;谷川好男,古屋淳;Masao Ishikawa and Hiroyuki Tagawa;Hiroyuki Yoshida;Y. Tanigawa;石川雅雄;Y.Tanigawa;Hiroyuki Yoshida;Jiang Zeng;Y. Tanigawa;Hiroyuki Yoshida;石川雅雄;Y.Tanigawa;梅田亨;石川雅雄;谷川好男,古屋淳;石川雅雄;古屋淳;石川雅雄;谷川好男;石川雅雄
• 通讯作者: 石川雅雄

A Partial Proof of Okada's (q ; t)-Hook Formula Conjecture for Birds

冈田 (q ; t)-Hook 公式鸟类猜想的部分证明

• 发表时间: 2010
• 作者: 谷川好男;古屋淳;石川雅雄;山田裕史;石川雅雄;山田裕史;石川雅雄
• 通讯作者: 石川雅雄