On the center of a Coxeter group

Abstract

In this paper, we show that the center of every Coxeter group is finite and isomorphic to $({\mathbb{Z}}_2{)}^n$ for some $n\ge 0$. Moreover, for a Coxeter system $(W,S)$, we prove that $Z(W)=Z(W_{S\setminus \tilde{S}})$ and $Z(W_{\tilde{S}})=1$, where $Z(W)$ is the center of the Coxeter group $W$ and $\tilde{S}$ is the subset of $S$ such that the parabolic subgroup $W_{\tilde{S}}$ is the {\it essential parabolic subgroup} of $(W,S)$ (i.e.\ $W_{\tilde{S}}$ is the minimum parabolic subgroup of finite index in $(W,S)$). The finiteness of the center of a Coxeter group implies that a splitting theorem holds for Coxeter groups.