The Poincar\'e Series of Relative Invariants of Finite Pseudo-Reflection Groups

DOI：10.3770/j.issn:1000-341X.2010.02.018

 作者 单位 南基洙 大连理工大学数学科学学院, 辽宁 大连 116024 秦小二 长江师范学院数学系, 重庆 408003

设$G$是作用在特征数为$0$的域$F$上的向量空间$V$的有限伪反射群,$\chi:G \longrightarrow F^{*}$是$G$的$1$-维表示,本文证明对$\forall g\in G,\chi(g)=(det g)^{\alpha}(0\leq\alpha\leq r-1)$, 其中$r$为$g$的阶. 另外指出了该群的相对不变式和一般不变式的关系,并根据一般不变式的Poincar\'e级数的Molien公式,计算出相对不变式的Poincar\'e级数.

Let $F$ be a field with characteristic $0$, $V=F^{n}$ the $n$-dimensional vector space over $F$ and let $G$ be a finite pseudo-reflection group which acts on $V$. Let $\chi :G\longrightarrow F^{\ast }$ be a $1$-dimensional representation of $G$. In this article we show that $\chi (g)=({\rm det}\,g)^{\alpha }(0\leq \alpha \leq r-1)$, where $g\in G$ and $r$ is the order of $g$. In addition, we characterize the relation between the relative invariants and the invariants of the group $G$, and then we use Molien's Theorem of invariants to compute the Poincar\'e series of relative invariants.