Definition 4

Let $I$ be a set. A Coxeter matrix of type $I$ is a symmetric square matrix $M = (m_{ij})_{i,j \in I}$ whose entries are integers or $+\infty$ satisfying the relations

A Coxeter graph of type $I$ is (by abuse of language) a pair consisting of a graph $\Gamma^3$ having $I$ as its set of vertices and a map $f$ from the set of edges of this graph to the set consisting of $+\infty$ and the set of integers $\geq 3$. $\Gamma$ is called the underlying graph of the Coxeter graph $(\Gamma, f)$.

A Coxeter graph $(\Gamma, f)$ is associated to any Coxeter matrix $M$ of type $I$ as follows:

the graph $\Gamma$ has set of vertices $I$ and set of edges the set pairs $\{i, j\}$ of elements of $I$ such that $m_{ij} \geq 3$, and the map $f$ associates to the edge $\{i, j\}$ the corresponding element $m_{ij}$ of $M$.

It is clear that this gives rise to a bijection between the set of Coxeter matrices of type $I$ and the set of Coxeter graphs of type $I$.

To assist the reader in following our arguments, we often represent a Coxeter graph of type $I$ by the diagram used to represent its underlying graph, and write either next to or above each edge $\{i, j\}$ the number $f(\{i, j\})$. We generally omit these numbers if they are equal to 3.

3 See the appendix for the definition and properties of graphs used here.

If (W, S) is a Coxeter system, the matrix $M = (m(s, s'))_{s, s' \in S}$, where $m(s, s')$ is the order of $ss'$, is a Coxeter matrix of type S which is called the Coxeter matrix of (W, S): indeed, $m(s, s) = 1$ since $s^2 = 1$ for all $s \in S$, and $m(s, s') = m(s', s) \geqslant 2$ if $s \neq s'$ since $ss' = (s's)^{-1}$ is then $\neq 1$. The Coxeter graph $(\Gamma, f)$ associated to M is called the Coxeter graph of (W, S). We remark that two vertices $s$ and $s'$ of $\Gamma$ are joined if and only if $s$ and $s'$ do not commute. For example, the Coxeter matrix of a dihedral group of order $2m$ is $\begin{pmatrix} 1 & m \\ m & 1 \end{pmatrix}$ and its Coxeter graph is represented by

when $m \geqslant 3$ (or

if $m = 3$) and by

when $m = 2$. \* The Coxeter graph of the symmetric group $\mathfrak{S}_n$ is represented by

($n - 1$ vertices.) \*

We show later (Chap. V, § 4) that, conversely, any Coxeter matrix is the matrix of a Coxeter system.

A Coxeter system (W, S) is said to be irreducible if the underlying graph of its Coxeter graph is connected (Appendix, no. 2) and non-empty. Equivalently, S is non-empty and there exists no partition of S into two distinct subsets S' and S'' of S such that every element of S' commutes with every element of S''. More generally, let $(\Gamma_i)_{i \in I}$ be the family of connected components of $\Gamma$ (Appendix, no. 2) and let $S_i$ be the set of vertices of $\Gamma_i$. Let $W_i = W_{S_i}$ be the subgroup of W generated by $S_i$ (cf. no. 8). Then the $(W_i, S_i)$ are irreducible Coxeter systems (no. 8, Th. 2) called the irreducible components of (W, S). Moreover, the group W is the restricted direct product[^1] of the subgroups $W_i$ for $i \in I$. Indeed, this follows from the following more general proposition: