Editing Coxeter group (section)

==Affine Coxeter groups==
[[File:Affine coxeter.svg|400px|thumb|Coxeter diagrams for the affine Coxeter groups]]
[[File:Stiefel diagram for G2.png|thumb|right|Stiefel diagram for the <math>G_2</math> root system]]
{{See also|Affine Dynkin diagram|Affine root system}}

The '''affine Coxeter groups''' form a second important series of Coxeter groups.  These are not finite themselves, but each contains a [[normal subgroup|normal]] [[abelian group|abelian]] [[subgroup]] such that the corresponding quotient group is finite.  In each case, the quotient group is itself a Coxeter group, and the Coxeter graph of the affine Coxeter group is obtained from the Coxeter graph of the quotient group by adding another vertex and one or two additional edges.  For example, for ''n''&nbsp;≥&nbsp;2, the graph consisting of ''n''+1 vertices in a circle is obtained from ''A<sub>n</sub>'' in this way, and the corresponding Coxeter group is the affine Weyl group of ''A<sub>n</sub>'' (the [[affine symmetric group]]).  For ''n''&nbsp;=&nbsp;2, this can be pictured as a subgroup of the symmetry group of the standard tiling of the plane by equilateral triangles.

In general, given a root system, one can construct the associated ''[[Eduard Stiefel|Stiefel]] diagram'', consisting of the hyperplanes orthogonal to the roots along with certain translates of these hyperplanes. The affine Coxeter group (or affine Weyl group) is then the group generated by the (affine) reflections about all the hyperplanes in the diagram.<ref>{{harvnb|Hall|2015}} Section 13.6</ref> The Stiefel diagram divides the plane into infinitely many connected components called ''alcoves'', and the affine Coxeter group acts freely and transitively on the alcoves, just as the ordinary Weyl group acts freely and transitively on the Weyl chambers. The figure at right illustrates the Stiefel diagram for the <math>G_2</math> root system.

Suppose <math>R</math> is an irreducible root system of rank <math>r>1</math> and let <math>\alpha_1,\ldots,\alpha_r</math> be a collection of simple roots. Let, also, <math>\alpha_{r+1}</math> denote the highest root. Then the affine Coxeter group is generated by the ordinary (linear) reflections about the hyperplanes perpendicular to <math>\alpha_1,\ldots,\alpha_r</math>, together with an affine reflection about a translate of the hyperplane perpendicular to <math>\alpha_{r+1}</math>. The Coxeter graph for the affine Weyl group is the Coxeter–Dynkin diagram for <math>R</math>, together with one additional node associated to <math>\alpha_{r+1}</math>. In this case, one alcove of the Stiefel diagram may be obtained by taking the fundamental Weyl chamber and cutting it by a translate of the hyperplane perpendicular to <math>\alpha_{r+1}</math>.<ref>{{harvnb|Hall|2015}} Chapter 13, Exercises 12 and 13</ref>

A list of the affine Coxeter groups follows:

{| class="wikitable"
!Group<br />symbol || [[Ernst Witt|Witt]]<br />symbol || Bracket notation|| Coxeter<br />graph || Related uniform tessellation(s) 
|- align=center
!<math>{\tilde{A}}_n</math> 
||<math>P_{n+1}</math> || [3<sup>[''n''+1]</sup>] || {{CDD|node|split1|nodes|3ab}}...{{CDD|3ab|nodes|3ab|branch}}<br />or<br />{{CDD|branch|3ab|nodes|3ab}}...{{CDD|3ab|nodes|3ab|branch}}|| [[Simplectic honeycomb]]
|- align=center
!<math>{\tilde{B}}_n</math> 
||<math>S_{n+1}</math> || [4,3<sup>''n'' − 3</sup>,3<sup>1,1</sup>]  || {{CDD|node|4|node|3|node|3}}...{{CDD|3|node|split1|nodes}}|| [[Demihypercubic honeycomb]]
|- align=center
!<math>{\tilde{C}}_n</math> 
||<math>R_{n+1}</math> || [4,3<sup>''n''−2</sup>,4]  || {{CDD|node|4|node|3|node|3}}...{{CDD|3|node|4|node}}|| [[Hypercubic honeycomb]]
|- align=center
!<math>{\tilde{D}}_n</math> 
||<math>Q_{n+1}</math> || [ 3<sup>1,1</sup>,3<sup>''n''−4</sup>,3<sup>1,1</sup>] || {{CDD|nodes|split2|node|3|node|3}}...{{CDD|3|node|split1|nodes}}||[[Demihypercubic honeycomb]]
|- align=center
!<math>{\tilde{E}}_6</math> 
||<math>T_{7}</math> || [3<sup>2,2,2</sup>] || {{CDD|nodea|3a|nodea|3a|branch|3ab|nodes|3a|nodea}} or {{CDD|nodes|3ab|nodes|split2|node|3|node|3|node}}|| [[2 22 honeycomb|2<sub>22</sub>]]
|- align=center
!<math>{\tilde{E}}_7</math> 
||<math>T_{8}</math> || [3<sup>3,3,1</sup>] || {{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}} or {{CDD|nodes|3ab||nodes|3ab|nodes|split2|node|3|node}}|| [[3 31 honeycomb|3<sub>31</sub>]], [[1 33 honeycomb|1<sub>33</sub>]]
|- align=center
!<math>{\tilde{E}}_8</math> 
||<math>T_{9}</math> || [3<sup>5,2,1</sup>] || {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}} || [[5 21 honeycomb|5<sub>21</sub>]], [[2 51 honeycomb|2<sub>51</sub>]], [[1 52 honeycomb|1<sub>52</sub>]]
|- align=center
!<math>{\tilde{F}}_4</math> 
||<math>U_{5}</math> || [3,4,3,3]|| {{CDD|node|3|node|4|node|3|node|3|node}} || [[16-cell honeycomb]]<br />[[24-cell honeycomb]]
|- align=center
!<math>{\tilde{G}}_2</math> 
||<math>V_{3}</math> || [6,3] || {{CDD|node|6|node|3|node}} || [[Hexagonal tiling]] and<br />[[Triangular tiling]]
|- align=center
! <math>{\tilde{A}}_1 = I_2(\infty)</math> 
||<math>W_{2}</math> || [∞] || {{CDD|node|infin|node}} || [[Apeirogon]]
|}

The group symbol subscript is one less than the number of nodes in each case, since each of these groups was obtained by adding a node to a finite group's graph.