Research on the holonomic q difference systems associated with the Jackson integrals of Weyl group invariant

与Weyl群不变量Jackson积分相关的完整q差分系统研究

• 批准号: 17540037
• 负责人: ITO Masahiko
• 金额: $ 2.28万
• 依托单位: Aoyama Gakuin University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540037/
• 关键词: Jackson integrals; Weyl group; q-difference system; basic hypergeometric series; connection formula; Vandermonde-type determinant; invariant polynomials of type F4; ルート系; Kazhdan-Lusztig多項式; 複素簡約型リー群; 対称式; ホロノミックq-差分方程式; カロジェロ模型

A key reason to consider the Jackson integrals, which admit Weyl group symmetry, is to provide an explanation and an extension of a series of classical basic hypergeometric series and their q-difference equations with respect to their parameters.In order to show the linear independence of the fundamental solutions of the q-difference system, the non-degeneracy of the Wronskian should be proved. For this purpose, Ito and K. Aomoto (Kyoto sangyo university) showed that the Wronskian is expressed as a product of q-gamma functions. Ito and Koike showed that a Vandermonde-type determinant, whose entries are the irreducible characters of the classical groups, is expressed as a power of the difference product. Technically speaking, the proof of the non-degeneracy of the Wronskian is obtained from the Vandermonde-type determinant, which is a limiting case of q-0 for the Wronskian. The formula given by Ito and Koike plays an important role in the result by Ito and Aomoto.Ito and Y. Sanada (Tsuda collage) gave an explicit connection formula, which indicates that the general solution of the q-difference system of the BC1-type Jackson integral is expressed as a linear combination of the fundamental solutions of the system. They showed that the connection formula is equivalent to the classical hypergeometric transformation formula, which was discovered in 1950s by Sears and Slater. Consequently they found the Weyl group symmetry in the transformation formula and gave a very simple proof for the formula. Ito also gave an explicit connection formula for the BCn-type Jackson integral by extending it from the result of Ito and Sanada.Taniguchi gave an elementary construction for the invariant polynomials of type F4 under the action of its Weyl group, using the degree 2 invariant polynomials under the action of Weyl group of type D4, based on the fact that Weyl group of type D4 is included in a normal subgroup of Weyl group of type F4.

考虑杰克逊积分的一个重要原因是为了解释和推广一系列经典的基本超几何级数及其q-差分方程关于它们的参数，为了证明q-差分方程组的基本解的线性无关性，必须证明Wronskian的非退化性。为此，Ito和K. Aomoto（京都产业大学）表明，朗斯基表示为产品的q-伽玛函数。伊藤和小池证明了一个范德蒙型行列式，其元素是经典群的不可约特征标，可以表示为差积的幂。从技术上讲，朗斯基的非简并性的证明是从范德蒙型行列式得到的，范德蒙型行列式是朗斯基的q-0的极限情况。Ito和Koike给出的公式在Ito和Aomoto的结果中起着重要的作用。Sanada（Tsuda Collage）给出了一个显式的联系公式，表明BC 1型杰克逊积分的q-差分系统的通解可表示为基本解的线性组合。他们证明了连接公式等价于Sears和斯莱特在20世纪50年代发现的经典超几何变换公式。因此，他们发现了变换公式中的Weyl群对称性，并给出了一个非常简单的公式证明。Ito推广了Ito和Sanada的结果，给出了BCn型杰克逊积分的一个显式联系公式，Taniguchi利用D4型Weyl群作用下的2次不变多项式，给出了F4型Weyl群作用下的不变多项式的一个初等构造，基于D_4型Weyl群包含在F_4型Weyl群的正规子群中这一事实，

项目成果

A generalization of Weyl's denominator formulas for the classical groups☆

• DOI: 10.1016/j.jalgebra.2006.05.026
• 发表时间: 2006-08
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Masahiko Ito;K. Koike

G2型マクドナルド対称多項式のピエリ公式

G2型McDonald对称多项式的Pieri公式

• 发表时间: 2006
• 作者: 伊藤 雅彦;真田 ゆかり;J.Komeda;伊藤 雅彦
• 通讯作者: 伊藤 雅彦

Another proof of Gustafson's C_n-type summation formula via 'elementary' symmetric polynomials

通过“初等”对称多项式证明 Gustafson C_n 型求和公式的另一个证明

• 期刊: Publications of Research Institute for Mathematical Sciences Kyoto University (To appear)
• 作者: Kazuhiko Aomoto;Masahiko Ito;Kenji Taniguchi;Tomomi Kawamura;Kenji TANIGUCHI;Tomomi KAWAMURA;Kenji Taniguchi;Masahiko Ito;Masahiko Ito;Masahiko ITO;Masahiko ITO;谷口 健二;Masahiko Ito;Masahiko Ito
• 通讯作者: Masahiko Ito

On the structure of Jackson integrals of B_n type and Holonomic q-difference equations

论B_n型Jackson积分与完整q-差分方程的结构

• 发表时间: 2005
• 期刊: Proceedings of the Japan Academy 81,Ser.A,No.9
• 作者: J.Komeda;A.Ohbuchi;K.Aomoto & M.Ito
• 通讯作者: K.Aomoto & M.Ito

a proof of transformation formula for a very-well-poised q-hypergeometric series from the view point of Weyl group symmetry

从Weyl群对称性的角度证明一个非常平衡的q-超几何级数的变换公式

• 发表时间: 2006
• 作者: Masahiko;Ito;Yukari;Sanada
• 通讯作者: Sanada