 
Summary: REFLECTION CENTRALIZERS IN COXETER GROUPS
DANIEL ALLCOCK
Abstract. We give a new proof of Brink's theorem that the non
reflection 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 com
puting 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 reflec
tion 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 Coxetergroup 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
