A Coxeter system is a group G with finite generating set S={s1, ..., sn}, defined by the power relations si2=1 for i=1, ..., n and braid relations
sisjsi ... = sjsisj ... for i, j=1, ..., n with i<j, where each side of this relation has length mij≥2. Although traditionally mij=∞ signifies that the corresponding relation is omitted, for technical reasons, we use mij=0 instead. Set mji=mij and mii=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, no distinction is made 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. Functions for manipulating these descriptions are described in Chapter COXETER SYSTEMS.
Coxeter groups are usually described by a Coxeter matrix M=(mij)i, j=1n, or by a Coxeter graph with vertices 1, ..., n and an edge connecting i and j labeled by mij whenever mij≥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, such a representation can be described 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, the naming system due to Cartan is also used. Hyperbolic Coxeter groups of degree larger than 3 are numbered.