where mi,j ≥ 2; the condition mi,j = ∞ means no relation of the form (xixj)m should be imposed. It is convenient to regard mi,j as a symmetric function of the indices i and j, which can then be encoded as a Coxeter graph in which the vertices stand for generator subscripts, i and j are connected if and only if mi,j ≥ 3, and the edge is labelled with the value of mi,j whenever it is 4 or greater. In particular, two generators commute if and only if they are not connected by an edge. This is because when x2 = y2 = 1, we have yx = xy if and only if xyxy = xxyy, i.e., if and only if (xy)2 = 1. In particular, if a Coxeter graph has two or more connected components, the associated group is the direct product of the groups associated to the individual components.
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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 Sn+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 Sn+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 An. 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 Bn and Cn give rise to the same Coxeter group. Also, G2 appears to be missing, but it is present under the name I6. The additions to the list are H3, H4, and the In.
All symmetry groups of regular polytopes are finite Coxeter groups. The dihedral groups, which are the symmetry groups of regular polygons, form the series In. The symmetry group of a regular n-simplex is the symmetric group Sn+1, also known as the Coxeter group of type An. The symmetry group of the n-dimensional hypercube is the same as that of the hyperoctahedron, namely BCn. The symmetry group of the regular dodecahedron and the regular icosahedron is H3. In dimension 4, there are three special regular polytopes, the 24-cell, the 120-cell, and the 600-cell. The first has symmetry group F4, while the other two have symmetry group H4.
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 An in this way, and the corresponding Coxeter group is the affine Weyl group of An. For n = 2, this can be pictured as the symmetry group of the standard tiling of the plane by equilateral triangles.
An example
Finite Coxeter groups
Symmetry groups of regular polytopes
Affine Weyl groups
Hyperbolic Coxeter groups
There are also hyperbolic Coxeter groups describing reflection groups in hyperbolic geometry.