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 Z W ðW I Þ of an arbitrary parabolic subgroup W I into the center of W I , 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 Z W ðW I Þ, 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.

On finite factors of centralizers of parabolic subgroups in Coxeter groups

“…The basics of Coxeter groups summarized here are found in [5] unless otherwise noticed. For some omitted definitions, see also [5] or the author's preceding paper [7].…”

“…This section summarizes some known properties (mainly proven in [7]) of the centralizers Z W (W I ) of parabolic subgroups W I in Coxeter groups W , especially those relevant to the argument in this paper. First, we fix an abstract index set Λ with |Λ| = |I|, and define S (Λ) to be the set of all injective mappings x : Λ → S. For x ∈ S (Λ) and λ ∈ Λ, we put x λ = x(λ); thus x may be regarded as a duplicate-free "Λ-tuple" (x λ ) = (x λ ) λ∈Λ of elements of S. For each x ∈ S (Λ) , let [x] denote the image of the mapping x; [x] = {x λ | λ ∈ Λ}.…”

“…Proposition 3.5 (see [7,Theorem 5.2]). If every irreducible component of I of finite type is of (−1)-type (see Section 2.3 for the terminology), then we have…”

“…An outline of the idea is as follows. First, by a general result on the structures of the centralizers of parabolic subgroups [7] or the normalizers of parabolic subgroups [2] in Coxeter groups applied to the case of a single reflection, we have a decomposition Z W (r) = r × (W ⊥r ⋊ Y r ), where W ⊥r denotes the subgroup generated by all the reflections except r itself that commute with r, and Y r is a subgroup isomorphic to the fundamental group of a certain graph associated to (W, S). The above-mentioned general results also give a canonical presentation of W ⊥r as a Coxeter group.…”

“…In Section 2, we summarize some fundamental properties and definitions for Coxeter groups. In Section 3, we summarize some properties of the centralizers of parabolic subgroups relevant to our argument in the following sections, which have been shown in some preceding works (mainly in [7]). In Section 4, we give the statement of the main theorem of this paper (Theorem 4.1), and give a remark on its application to the isomorphism problem in Coxeter groups (also mentioned in a paragraph above).…”