Coxeter matrix

The Coxeter matrix of a Coxeter system is the symmetric matrix whose entry $m_{ij}$ is the order of $x_i{x}_j$.

The Coxeter graph has vertex set $I$, joins $i$ and $j$ when $m_{ij}\ge 3$, and labels the edge by $m_{ij}$ when $m_{ij}>3$. A Coxeter system is irreducible exactly when this graph is connected.

An irreducible Coxeter system is one whose Coxeter graph is connected. The connected components of a Coxeter graph give the direct-product decomposition of its Coxeter group.

The type $E(p,q,k)$ graph is a simply-laced tree with one trivalent vertex and arms containing $p$, $q$, and $k$ vertices beyond that vertex.