Reflection group

Examples

Two dimensions

In two dimensions, the finite reflection groups are the dihedral groups, which are generated by reflection in two lines that form an angle of ${\displaystyle 2\pi /n}$ and correspond to the Coxeter diagram ${\displaystyle I_2(n).}$ Conversely, the cyclic point groups in two dimensions are not generated by reflections, nor contain any – they are subgroups of index 2 of a dihedral group.

Infinite reflection groups include the frieze groups ${\displaystyle \ast \mathrm{\infty }\mathrm{\infty }}$ and ${\displaystyle \ast 22\mathrm{\infty }}$ and the wallpaper groups ${\displaystyle \ast \ast }$, ${\displaystyle \ast 2222}$, ${\displaystyle \ast 333}$, ${\displaystyle \ast 442}$ and ${\displaystyle \ast 632}$. If the angle between two lines is an irrational multiple of pi, the group generated by reflections in these lines is infinite and non-discrete, hence, it is not a reflection group.

Three dimensions

Finite reflection groups are the point groups C_nv, D_nh, and the symmetry groups of the five Platonic solids. Dual regular polyhedra (cube and octahedron, as well as dodecahedron and icosahedron) give rise to isomorphic symmetry groups. The classification of finite reflection groups of R³ is an instance of the ADE classification.

Relation with Coxeter groups

A reflection group W admits a presentation of a special kind discovered and studied by H. S. M. Coxeter. The reflections in the faces of a fixed fundamental "chamber" are generators r_i of W of order 2. All relations between them formally follow from the relations

${\displaystyle (r_i{r}_j{)}^{c_{ij}}=1,}$

expressing the fact that the product of the reflections r_i and r_j in two hyperplanes H_i and H_j meeting at an angle ${\displaystyle \pi /c_{ij}}$ is a rotation by the angle ${\displaystyle 2\pi /c_{ij}}$ fixing the subspace H_i ∩ H_j of codimension 2. Thus, viewed as an abstract group, every reflection group is a Coxeter group.

Finite fields

When working over finite fields, one defines a "reflection" as a map that fixes a hyperplane. Geometrically, this amounts to including shears in a hyperplane. Reflection groups over finite fields of characteristic not 2 were classified by Zalesskiĭ & Serežkin (1981).

Generalizations

Discrete isometry groups of more general Riemannian manifolds generated by reflections have also been considered. The most important class arises from Riemannian symmetric spaces of rank 1: the n-sphere Sⁿ, corresponding to finite reflection groups, the Euclidean space Rⁿ, corresponding to affine reflection groups, and the hyperbolic space Hⁿ, where the corresponding groups are called hyperbolic reflection groups. In two dimensions, triangle groups include reflection groups of all three kinds.