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

基本信息

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

项目摘要

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.

考虑承认 Weyl 群对称性的 Jackson 积分的一个关键原因是提供一系列经典基本超几何级数及其 q 差分方程关于其参数的解释和扩展。为了显示线性独立性q-差分系统的基本解中，应证明朗斯基式的非简并性。为此，Ito 和 K. Aomoto（京都产业大学）表明，Wronskian 表示为 q-gamma 函数的乘积。伊藤和小池证明，范德蒙型行列式的条目是经典群的不可约特征，被表示为差积的幂。从技术上讲，朗斯基式非简并性的证明是从范德蒙型行列式获得的，它是朗斯基式q-0的极限情况。 Ito 和 Koike 给出的公式在 Ito 和 Aomoto 的结果中起着重要作用。Ito 和 Y. Sanada（津田大学）给出了一个显式的连接公式，这表明 BC1- 的 q 差分系统的通解Jackson 型积分表示为系统基本解的线性组合。他们证明连接公式等价于西尔斯和斯莱特在 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，并给出了该公式的非常简单的证明。伊藤还通过对伊藤和真田结果的扩展，给出了BCn型杰克逊积分的显式联系公式。谷口利用2次不变量给出了在其Weyl群作用下的F4型不变量多项式的初等构造D4型Weyl群作用下的多项式，基于D4型Weyl群包含在F4型Weyl群的正规子群中的事实。