Coxeter group
A Coxeter group is a mathematical group generated by reflections in Euclidean space (or, more abstractly, any group isomorphic to such a reflection group). Coxeter groups are ubiquitous in mathematics and geometry. The finite Coxeter groups are precisely the finite Euclidean reflection groups. Consequently, the symmetry groups of all the regular polytopes are finite Coxeter groups. The Weyl groups of root systems are also all special cases of finite Coxeter groups. A triangle group is a special kind of Coxeter group, generated by reflections in the sides of a triangle in the sphere, the Euclidean plane, or the hyperbolic plane.
Coxeter groups are named for the geometer H. S. M. Coxeter.
Contents
Definition
Formally, a Coxeter group can be defined as a group with the presentation
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle r_1,r_2,\ldots,r_n \mid (r_ir_j)^{m_{ij}}=1\right\rangle}
where m_{ii} = 1 and m_{ij} ≥ 2 for i ≠ j. The condition m_{ij} = ∞ means no relation of the form (r_{i} r_{j})^{m} should be imposed. The relation m_{ii} = 1 means that (r_{i})^{2} = 1 for all i (i.e. the generators are involutions). Under the condition that (r_{i})^{2} = 1 one can show that
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (r_ir_j)^{m_{ij}} = 1 \;\Leftrightarrow\; (r_jr_i)^{m_{ij}} = 1}
It is then convenient to regard m_{ij} as the entries of a n×n symmetric matrix with 1's on the diagonal called the Coxeter matrix.
The Coxeter matrix can be encoded as a Coxeter graph in which the vertices stand for generator subscripts, i and j are connected if and only if m_{ij} ≥ 3, and the edge is labelled with the value of m_{ij} 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
- x^{2} = y^{2} = 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, G_{2} appears to be missing, but it is present under the name I_{2}(6). The additions to the list are H_{3}, H_{4}, and the I_{2}(p).
Some properties of the finite Coxeter groups are given in the following table:
Type | Rank | Order | Polytope |
---|---|---|---|
A_{n} | n | (n + 1)! | n-simplex |
B_{n} = C_{n} | n | 2^{n} n! | n-cube / n-cross-polytope |
D_{n} | n | 2^{n−1} n! | |
I_{2}(p) | 2 | 2p | p-gon |
H_{3} | 3 | 120 | icosahedron / dodecahedron |
F_{4} | 4 | 1152 | 24-cell |
H_{4} | 4 | 14400 | 120-cell / 600-cell |
E_{6} | 6 | 51840 | E_{6} polytope |
E_{7} | 7 | 2903040 | |
E_{8} | 8 | 696729600 |
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_{2}(p). 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 H_{3}. In dimension 4, there are three special regular polytopes, the 24-cell, the 120-cell, and the 600-cell. The first has symmetry group F_{4}, while the other two have symmetry group H_{4}.
The Coxeter groups of type D_{n}, E_{6}, E_{7}, and E_{8} are the symmetry groups of certain semiregular polytopes.
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.