Basic sets of invariants for finite reflection groups
Basic sets of invariants for finite reflection groups
复制标题
有限反射群的基本不变量集
DOI：
10.1090/s0002-9904-1968-12017-8
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发表时间：
1968
影响因子：
1.3
通讯作者：
L. Flatto
中科院分区：
文献类型：
--
作者：
L. Flatto
1. 介绍。设V是特征为0的域K上的w维向量空间。设G是v的线性变换的有限群，如果我们定义（gP）(x)=-P （G ~ 1#）对于gÇzG， P (x)(EK [x])，则G自然是多项式环K [x]的自同构群。在G下不变的多项式形成了一个I / K的代数，称为G的不变量代数。如果一个线性变换有有限阶，并且留下一个固定的（»-l）维超平面，称为它的反射超平面，那么它就是一个反射。如果G是有限阶且由反射产生，则G是有限反射群。Chevalley[5]证明了对于有限反射群，J有一个由n个代数独立形式ii，•••，In组成的完整基。此外，Shephard和Todd [lO]已经证明了有限反射群的这种性质。如果K是实数域R，则G留下不变的正定二次型[2]，使得G在变量线性变化后是正交的。Coxeter[3]，[4]对所有不可约有限正交反射群进行了分类，并计算了形式ii， 72，•••，In*的度mi，•••，mn，这些度与特定选择的基无关。我们提供了一种计算这些群的I的显式完整性基的方法。我们将把这个问题与某个均值问题联系起来。
1. Introduction. Let V be an w-dimensional vector space over a field K of characteristic zero. Let G be a finite group of linear transformations of V. Gacts naturally as a group of automorphisms of the ring of polynomials K [x] if we define (gP)(x)=-P (g~ 1#) for gÇzG, P (x)(EK [x]. The polynomials which are invariant under G form an algebra I over K called the algebra of invariants of G. A linear transformation is said to be a reflection if it has finite order and leaves fixed an (»—l)-dimensional hyperplane, called its reflecting hyperplane. G is a finite reflection group if it is of finite order and is generated by reflections. Chevalley [5] has proved that for finite reflection groups, J has an integrity basis consisting of n algebraically independent forms ii,•••, In. Furthermore, Shephard and Todd [lO] have shown that this property of I characterizes the finite reflection groups.If K is the real field R, then G leaves invariant a positive definite quadratic form [2] so that G is orthogonal after a linear change of variables. Coxeter [3],[4] has classified all irreducible finite orthogonal reflection groups and has computed the degrees mi,•••, mn of the forms ii, 72,•••, In* These degrees are independent of the particularly chosen basis. We provide a method for computing an explicit integrity basis of I for these groups. We will relate this problem to a certain mean value problem.