Reflection Centralizers in Coxeter Groups

We give a new proof of Brink’s theorem that the nonreflection part of a reflection centralizer in a Coxeter group is free, and make several refinements. In particular we give an explicit finite set of generators for the centralizer and a method for computing the Coxeter diagram for its reflection part. In many cases, our method allows one to compute centralizers quickly in one’s head. We also define “Vinberg representations” of Coxeter groups, in order to isolate some of the key properties of the Tits cone. Brink has proved the elegant result that the centralizer of a reflection in a Coxeter group is the semidirect product of a Coxeter group by a free group [5]. In fact this free group is the fundamental group of the component of the “odd Coxeter diagram” distinguished by the conjugacy class of the reflection. We give a new proof of her result, together with several refinements. The first refinement is a method of computing the Coxeter diagram of the Coxeter-group part of the centralizer. With a little effort we develop this method to the point that many centralizer computations are very easy. For example, the fact that the reflection centralizer in W (E8) is W (E7) × 2 becomes an almost-instant mental computation. We offer many other examples, including the reflection centralizer in the Coxeter group of Bugaenko that acts cocompactly on 8-dimensional hyperbolic space [7]. Our method shares the same foundation as that of Brink and Howlett [6], which is a special case of an algorithm for understanding normalizers of parabolic subgroups. (See also [1] and [3] for related work.) However, in use it feels quite different. The second refinement is an explicit finite set of generators for the reflection centralizer; Brink only gave explicit generators for the free part. This generating set plays a key role in the author’s work [2] with Lisa Carbone on Kac-Moody groups. Our proof of Brink’s theorem is quite different from hers, using covering spaces and topology in place of induction on word lengths. We hope this alternate proof will be helpful to some people. In order to Date: June 20, 2011. 2000 Mathematics Subject Classification. 20F55.