On the center of a Coxeter group

Original · EN

In this paper, we show that the center of every Coxeter group is finite and isomorphic to (₂)ⁿ for some n≥ 0. Moreover, for a Coxeter system (W,S), we prove that Z(W)=Z(WSS) and Z(WS)=1, where Z(W) is the center of the Coxeter group W and S is the subset of S such that the parabolic subgroup WS is the essential parabolic subgroup of (W,S) (i.e.WS 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.