Introduction

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

\[s_is_js_i\cdots = s_js_is_j\cdots\]

for \(i,j=1,\dots,n\) with \(i<j\), where each side of this relation has length \(m_{ij}\ge 2\). Traditionally, \(m_{ij}=\infty\) signifies that the corresponding relation is omitted but for technical reasons \(m_{ij}=0\) is used in Magma 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, no distinction is made between a Coxeter system and its Coxeter group. See [Bourbaki, 1968] for more details on the theory of Coxeter groups.

The rank of the Coxeter system is \(n=|S|\). A Coxeter system is said to be reducible if there is a proper subset \(I\) of \(\{1,\dots,n\}\) 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 \(W_I=\langle s_i \mid i \in I \rangle\) and \(W_{I^c}=\langle s_i \mid i \notin I \rangle\). Note that an irreducible Coxeter group may still be a nontrivial direct product of abstract subgroups (for example, \(W(G_2)\cong S_2\times S_3\)). Two Coxeter systems are Coxeter isomorphic (or graph 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, Coxeter matrices, Coxeter graphs, Cartan matrices, and Dynkin digraphs will be discussed. The classification of finite and affine Coxeter groups provides a naming system for these groups. In Chapters ChapRootSys and ChapRootDtm, finite root systems and root data, which provide a more detailed description of finite Coxeter groups, are discussed. Coxeter groups themselves are discussed in Chapter ChapGrpCox; reflection representations of Coxeter groups are discussed in Chapter ChapGrpRfl.