The Poincar\'e Series of Relative Invariants of Finite Pseudo-Reflection Groups
Received:April 26, 2008  Revised:October 06, 2008
Key Word: Poincar\'e series   finite pseudo-reflection group   relative invariants.
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.10771023).
 Author Name Affiliation Ji Zhu NAN School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China Xiao Er QIN Department of Mathematics, Yangtze Normal University, Chongqing 408003, P. R. China
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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.