On centralizers of parabolic subgroups in Coxeter groups

Abstract

Abstract Let W be an arbitrary Coxeter group, possibly of infinite rank. We describe a decomposition of the centralizer ZW (WI ) of an arbitrary parabolic subgroup WI into the center of WI , a Coxeter group and a subgroup defined by a 2-cell complex. Only information about finite parabolic subgroups is required in an explicit computation. By using our description of ZW (WI ), we will be able to reveal a further strong property of the action of the third factor on the second factor, in particular on the finite irreducible components of the second factor.