coxeter group

In mathematics, a Coxeter group is a group with a presentation of the form

\left\langle x_1,x_2,\ldots,x_n|x_1^2=x_2^2=\ldots =x_n^2 =1,(x_1x_2)^{m_{1,2}}=(x_1x_3)^{m_{1,3}}=\ldots =(x_{n-1}x_n)^{m_{n-1,n}}=1\right\rangle

where m_i,j ≥ 2; the condition m_i,j = ∞ means no relation of the form (x_ix_j)^m should be imposed. It is convenient to regard m_i,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 m_i,j ≥ 3, and the edge is labelled with the value of m_i,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.

An example

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.

Finite Coxeter groups

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 C_n 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 I_n.

Symmetry groups of regular polytopes

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 n-simplex is the symmetric group S_n+1, also known as the Coxeter group of type A_n. The symmetry group of the n-cube is the same as that of the n-cross-polytope, namely BC_n. 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.

Affine Weyl groups

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 A_n. For n = 2, this can be pictured as the symmetry group of the standard tiling of the plane by equilateral triangles.

Hyperbolic Coxeter groups

There are also hyperbolic Coxeter groups describing reflection groups in hyperbolic geometry.

Coxeter Group

An example

Finite Coxeter groups

Symmetry groups of regular polytopes

Affine Weyl groups

Hyperbolic Coxeter groups

See Also