On centralizers of parabolic subgroups in Coxeter groups

Abstract

Let $W$ be an arbitrary Coxeter group, possibly of infinite rank. We describe a decomposition of the centralizer $Z_W(W_I)$ 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. Moreover, by using our description of $Z_W(W_I)$, we 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.

33 pages; improved version, notations changed (ver. 2); inappropriate Eq. (4.7) was removed (ver. 3); Theorem 7.1 was improved (ver. 4); the content of Section 7 separated as an individual paper and removed from this paper (ver. 5)

References in corpus (4)

• Irreducible Coxeter groups
• On the direct indecomposability of infinite irreducible Coxeter groups and the Isomorphism Problem of Coxeter groups
• Commensurators of parabolic subgroups of Coxeter groups
• Centralizers of reflections and reflection-independence of Coxeter groups

Cited by in corpus (5)

• Centralizers of reflections and reflection-independence of Coxeter groups
• Locally parabolic subgroups in Coxeter groups of arbitrary ranks
• Almost central involutions in split extensions of Coxeter groups by graph automorphisms
• Poincaré series of double coset representatives of Coxeter groups
• On finite factors of centralizers of parabolic subgroups in Coxeter groups