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

Abstract:

      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.