Reflection group

In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections (by the Cartan–Dieudonné theorem), it is a continuous group (indeed, Lie group), not a discrete group, and is generally considered separately. In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections (by the Cartan–Dieudonné theorem), it is a continuous group (indeed, Lie group), not a discrete group, and is generally considered separately. Let E be a finite-dimensional Euclidean space. A finite reflection group is a subgroup of the general linear group of E which is generated by a set of orthogonal reflections across hyperplanes passing through the origin. An affine reflection group is a discrete subgroup of the affine group of E that is generated by a set of affine reflections of E (without the requirement that the reflection hyperplanes pass through the origin). The corresponding notions can be defined over other fields, leading to complex reflection groups and analogues of reflection groups over a finite field. In two dimensions, the finite reflection groups are the dihedral groups, which are generated by reflection in two lines that form an angle of 2 π / n {displaystyle 2pi /n} and correspond to the Coxeter diagram I 2 ( n ) . {displaystyle I_{2}(n).} Conversely, the cyclic point groups in two dimensions are not generated by reflections, and indeed contain no reflections – they are however subgroups of index 2 of a dihedral group. Infinite reflection groups include the frieze groups ∗ ∞ ∞ {displaystyle *infty infty } and ∗ 22 ∞ {displaystyle *22infty } and the wallpaper groups ∗ ∗ {displaystyle **} , ∗ 2222 {displaystyle *2222} , ∗ 333 {displaystyle *333} , ∗ 442 {displaystyle *442} and ∗ 632 {displaystyle *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. Finite reflection groups are the point groups Cnv, Dnh, 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. Reflection groups have deep relations with kaleidoscopes, as discussed in (Goodman 2004). 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 ri of W of order 2. All relations between them formally follow from the relations expressing the fact that the product of the reflections ri and rj in two hyperplanes Hi and Hj meeting at an angle π / c i j {displaystyle pi /c_{ij}} is a rotation by the angle 2 π / c i j {displaystyle 2pi /c_{ij}} fixing the subspace Hi ∩ Hj of codimension 2. Thus, viewed as an abstract group, every reflection group is a Coxeter group.