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

考虑承认Weyl群对称的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关于其参数的q-差分方程的解释和扩展。为了证明q差分系统基本解的线性无关性，必须证明朗斯基行列式的非简并性。为此，Ito和K. Aomoto（京都工业大学）证明了朗斯基行列式被表示为q-函数的乘积。Ito和Koike证明了vandermonde型行列式的项是经典群的不可约特征，它被表示为差积的幂。从技术上讲，朗斯基行列式的非简并性的证明是由vandermonde型行列式得到的，它是朗斯基行列式q-0的极限情况。伊藤和小池给出的公式在伊藤和青本的结果中发挥了重要作用。Ito和Y. Sanada （Tsuda collage）给出了一个显式连接公式，该公式表明bc1型Jackson积分的q差分系统的通解表示为该系统的基本解的线性组合。他们证明了连接公式等价于经典的超几何变换公式，后者是由Sears和Slater在20世纪50年代发现的。因此，他们在变换公式中发现了Weyl群对称，并给出了一个非常简单的证明。Ito还将Ito和Sanada的结果推广到bcn型Jackson积分，给出了bcn型Jackson积分的显式连接公式。Taniguchi基于D4型Weyl群包含在F4型Weyl群的正规子群中的事实，利用D4型Weyl群作用下的2次不变多项式，给出了其Weyl群作用下F4型不变多项式的初等构造。

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.