Introduction

The functions in this chapter handle basic descriptions of Coxeter systems. A Coxeter system is a group G together with finite generating set S={s_1, ..., s_n}, and relations s_i^2=1 for i=1, ..., n and

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. 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. See [Bou68] for more details on the theory of Coxeter groups.

The rank of the Coxeter system is n=|S|. We say that a Coxeter system is reducible if there is a proper subset I of {1, ..., n} such that m_(ij)=2 or m_(ji)=2 whenever i in I and j notin I. In this case, G is an (internal) direct product of the Coxeter subgroups W_I=< s_i | i in I > and W_(I^c)=< s_i | i notin I >. Note that an irreducible Coxeter group may still be a nontrivial direct product of abstract subgroups (for example, W(G_2) isomorphic to S_2 x S_3). Two Coxeter groups are Coxeter isomorphic if there is a group isomorphism between them which takes Coxeter generators to Coxeter generators. In other words, the two groups are the same modulo renumbering of the generators.

Coxeter groups and their representations as reflection groups have a number of useful descriptions. In this chapter, we discuss Coxeter matrices, Coxeter graphs, Cartan matrices, and Dynkin digraphs. We also give the classification of finite and affine Coxeter groups, which provides a naming system for these groups. In Chapters ROOT SYSTEMS and ROOT DATA we discuss finite root systems and root data, which provide a more detailed description of finite Coxeter groups. Coxeter groups themselves are discussed in Chapter COXETER GROUPS; finite Coxeter groups as permutation groups are discussed in Chapter COXETER GROUPS AS PERMUTATION GROUPS; reflection representations of Coxeter groups are discussed in Chapter REFLECTION GROUPS.