On finite factors of centralizers of parabolic subgroups in Coxeter groups

Abstract

It has been known that the centralizer Z_W(W_I) of a parabolic subgroup W_I of a Coxeter group W is a split extension of a naturally defined reflection subgroup by a subgroup defined by a 2-cell complex $\mathscr{Y}$. In this paper, we study the structure of Z_W(W_I) further and show that, if I has no irreducible components of type A_n with 2 ≤ n < ∞, then every element of finite irreducible components of the inner factor is fixed by a natural action of the fundamental group of $\mathscr{Y}$. This property has an application to the isomorphism problem in Coxeter groups.