Descriptions of Coxeter Groups

A Coxeter system is a group G together with finite generating set S={s_1, ..., s_n}, relations s_i^2=1 for i=1, ..., n and braid relations

s_is_js_i ... = s_js_is_j ... for i, j=1, ..., n with i<j, where each side of this relation has length m_(ij) >= 2. Traditionally m_(ij)=Infinity signifies that the corresponding relation is omitted---for technical reasons, we use m_(ij)=0 instead. Set m_(ji)=m_(ij) and m_(ii)=1. The group G is called a Coxeter group and S is called the set of Coxeter generators. Since every group in Magma has a preferred generating set, we make no distinction between a Coxeter system and its Coxeter group.

Due to the importance and ubiquity of Coxeter groups, a number of different ways of describing these groups and their reflections have been developed. We provide functions for manipulating these descriptions in Chapter COXETER SYSTEMS.

Coxeter groups are usually described by a Coxeter matrix M=(m_(ij))_(i, j=1)^n, or by a Coxeter graph with vertices 1, ..., n and an edge connecting i and j labeled by m_(ij) whenever m_(ij) >= 3.

Coxeter systems are mainly important because they provide presentations for the real reflection groups. A Cartan matrix describes a particular reflection representation of a Coxeter group. In certain cases, we can also describe such a representation by an integer labelled digraph, called the Dynkin digraph (this is equivalent to a Dynkin diagram, but we have modified the definition for technical reasons).

For finite and affine Coxeter groups, we also use the naming system due to Cartan. Hyperbolic Coxeter groups of degree larger than 3 are numbered.