where *m*_{i,j} ≥ 2; the condition *m*_{i,j} = ∞ means no relation of the form (*x*_{i}*x*_{j})^{m} should be imposed. It is convenient to regard *m _{i,j}* as a symmetric function of the indices

Table of contents |

2 Finite Coxeter groups 3 Symmetry groups of regular polytopes 4 Affine Weyl groups 5 Hyperbolic Coxeter groups |

The graph in which vertices 1 through *n* are placed in a row with each vertex connected by an unlabelled edge to its immediate neighbors gives rise to the symmetric group *S*_{n+1}; the generators correspond to the transpositions (1 2), (2 3), ... (*n* *n*+1). Two non-consecutive transpositions always commute, while (*k* *k*+1) (*k*+1 *k\*+2) gives the 3-cycle (*k* *k*+1 *k*+2). Of course this only shows that *S _{n+1}* is a quotient group of the Coxeter group, but it is not too difficult to check that equality holds.

Every Weyl group can be realized as a Coxeter group. The Coxeter graph can be obtained from the Dynkin diagram by replacing every double edge with an edge labelled 4 and every triple edge by an edge labelled 6. The example given above corresponds to the Weyl group of the root system of type *A _{n}*. The Weyl groups include most of the finite Coxeter groups, but there are additional examples as well. The following list gives all connected Coxeter graphs giving rise to finite groups:

Comparing this with the list of simple root systems, we see that *B _{n}* and

All symmetry groups of regular polytopes are finite Coxeter groups. The dihedral groups, which are the symmetry groups of regular polygons, form the series *I _{n}*. The symmetry group of a regular

The affine Weyl groups form a second important series of Coxeter groups. These are not finite themselves, but each contains a normal abelian subgroup such that the corresponding quotient group is finite. In each case, the quotient group is itself a Weyl group, and the Coxeter graph is obtained from the Coxeter graph of the Weyl group by adding an additional vertex and one or two additional edges. For example, for *n* ≥ 2, the graph consisting of *n*+1 vertices in a circle is obtained from *A _{n}* in this way, and the corresponding Coxeter group is the affine Weyl group of