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arXiv 0810.0518math.CAmath.COmath.GR
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4F3（1）超几何级数上的Coxeter群作用

Coxeter group actions on 4F3(1) hypergeometric series

Marc Formichella, R. M. Green, Eric Stade

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30 pages, AMSTeX
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摘要：

研究了两个${_4}F_3(1)$型Saalschutzian超几何级数的线性组合$K(\vec{x})=K(a;b,c,d;e,f,g)$.我们首先证明了$K(a;b,c,d;e,f,g)$在某个矩阵群$G_K$的作用下是不变的，该矩阵群同构于对称群$S_6$，作用在仿射超平面$V=\{(a,b,c,d,e,f,g)\in\Bbb C^7\colon e+f+g-a-b-c-d=1\}$上。我们进一步发展了$K(a;b,c,d;e,f,g)$的三项关系的代数。本文证明了，对于同构于Coxeter群$W(D_6)$（阶数为23040）并包含上述群$G_K$的某个矩阵群$M_K$的任意三个元素$μ_1,μ_2,μ_3$，只要$μ_j$中没有两个在$M_K$中的$G_K$的同一右陪集上，则$K(μ_1\vec{x})$，$K(μ_2\vec{x})$和$K(μ_3\vec{x})$之间存在关系.这些三项关系中的系数被视为$a,b,c,d,e,f,g$中伽马和正弦函数的有理组合。根据所讨论的三元组$(μ_1,μ_2,μ_3) $的汉明类型，由$({|M_K|/|G_K|\atop 3})=({32\atop 3})=4960$产生的三项关系的集合可以进一步被划分为五个子集。该汉明类型根据$μ_j$之间的汉明距离来定义，而汉明距离又根据作为Coxeter群生成元中的字的$μ_j$的表达式来定义.每一个给定的汉明类型的三项关系都可以通过变量的改变而变换成任何其他相同类型的关系。提供了五种类型的三项关系中的每一种的明确示例。

英文摘要：

We investigate a certain linear combination $K(\vec{x})=K(a;b,c,d;e,f,g)$ of two Saalschutzian hypergeometric series of type ${_4}F_3(1)$. We first show that $K(a;b,c,d;e,f,g)$ is invariant under the action of a certain matrix group $G_K$, isomorphic to the symmetric group $S_6$, acting on the affine hyperplane $V=\{(a,b,c,d,e,f,g)\in\Bbb C^7\colon e+f+g-a-b-c-d=1\}$. We further develop an algebra of three-term relations for $K(a;b,c,d;e,f,g)$. We show that, for any three elements $μ_1,μ_2,μ_3$ of a certain matrix group $M_K$, isomorphic to the Coxeter group $W(D_6)$ (of order 23040), and containing the above group $G_K$, there is a relation among $K(μ_1\vec{x})$, $K(μ_2\vec{x})$, and $K(μ_3\vec{x})$, provided no two of the $μ_j$'s are in the same right coset of $G_K$ in $M_K$. The coefficients in these three-term relations are seen to be rational combinations of gamma and sine functions in $a,b,c,d,e,f,g$. The set of $({|M_K|/|G_K|\atop 3})=({32\atop 3})=4960$ resulting three-term relations may further be partitioned into five subsets, according to the Hamming type of the triple $(μ_1,μ_2,μ_3) $ in question. This Hamming type is defined in terms of Hamming distance between the $μ_j$'s, which in turn is defined in terms of the expression of the $μ_j$'s as words in the Coxeter group generators. Each three-term relation of a given Hamming type may be transformed into any other of the same type by a change of variable. An explicit example of each of the five types of three-term relations is provided.

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