Editing Coxeter group

{{Short description|Group that admits a formal description in terms of reflections}}
In [[mathematics]], a '''Coxeter group''', named after [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]], is an [[group (mathematics)|abstract group]] that admits a  [[Presentation of a group|formal description]] in terms of [[Reflection (mathematics)|reflections]] (or [[Kaleidoscope|kaleidoscopic mirrors]]). Indeed, the finite Coxeter groups are precisely the finite Euclidean [[reflection group]]s; for example, the [[symmetry group]] of each [[regular polyhedron]] is a finite Coxeter group.  However, not all Coxeter groups are finite, and not all can be described in terms of [[Symmetry in mathematics|symmetries]] and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups,<ref name="Coxeter1934">{{cite journal|title=Discrete groups generated by reflections|last=Coxeter|first=H. S. M.|journal=Annals of Mathematics|volume=35|pages=588–621 |year=1934|issue=3 |citeseerx=10.1.1.128.471|doi=10.2307/1968753|jstor=1968753|language=en}}</ref> and finite Coxeter groups were classified in 1935.<ref name="Coxeter1935">{{cite journal |title=The complete enumeration of finite groups of the form <math>r_{i}^{2}=(r_{i}r_{j})^{k_{ij}}=1</math>|last=Coxeter|first=H. S. M. |journal=Journal of the London Mathematical Society |pages=21–25 |date=January 1935|language=en|doi=10.1112/jlms/s1-10.37.21}}</ref>

Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of  [[regular polytope]]s, and the [[Weyl group]]s of [[simple Lie algebra]]s.  Examples of infinite Coxeter groups include the [[triangle group]]s corresponding to [[Tessellation#Overview|regular tessellation]]s of the [[Euclidean plane]] and the [[Hyperbolic space|hyperbolic plane]], and the Weyl groups of infinite-dimensional [[Kac–Moody algebra]]s.<ref>{{cite book|title=Lie Groups and Lie Algebras|last=Bourbaki|first=Nicolas|year=2002|chapter=4-6|publisher=Springer|language=en|isbn=978-3-540-42650-9|zbl=0983.17001|series=Elements of Mathematics}}</ref><ref>{{cite book|title=Reflection Groups and Coxeter Groups|last=Humphreys|first=James E.|url=https://sites.math.washington.edu/~billey/classes/reflection.groups/references/Humphreys.ReflectionGroupsAndCoxeterGroups.pdf|series=Cambridge Studies in Advanced Mathematics|volume=29|access-date=2023-11-18|year=1990|publisher=Cambridge University Press|doi=10.1017/CBO9780511623646|isbn=978-0-521-43613-7|zbl=0725.20028}}</ref><ref>{{cite book|title=The Geometry and Topology of Coxeter Groups|last=Davis|first=Michael W.|url=https://people.math.osu.edu/davis.12/davisbook.pdf|access-date=2023-11-18|year=2007|publisher=Princeton University Press|language=en|isbn=978-0-691-13138-2|zbl=1142.20020}}</ref>

==Definition==
Formally, a '''Coxeter group''' can be defined as a group with the [[Presentation of a group|presentation]]

:<math>\left\langle r_1,r_2,\ldots,r_n \mid (r_ir_j)^{m_{ij}}=1\right\rangle</math>

where <math>m_{ii}=1</math> and <math>m_{ij} = m_{ji} \ge 2</math> is either an integer or <math> \infty </math> for <math>i\neq j</math>.
Here, the condition <math>m_{i j}=\infty</math>  means that no relation of the form <math>(r_ir_j)^m = 1</math> for any integer <math> m \ge 2</math> should be imposed.

The pair <math>(W,S)</math> where <math>W</math> is a Coxeter group with generators <math>S=\{r_1, \dots , r_n\}</math> is called a '''Coxeter system'''. Note that in general <math>S</math> is ''not'' uniquely determined by <math>W</math>. For example, the Coxeter groups of type <math>B_3</math> and <math>A_1\times A_3</math> are isomorphic but the Coxeter systems are not equivalent, since the former has 3 generators and the latter has 1 + 3 = 4 generators (see below for an explanation of this notation).

A number of conclusions can be drawn immediately from the above definition.
* The relation <math>m_{ii} = 1</math> means that <math>(r_ir_i)^1 = (r_i)^2 = 1</math> for all <math>i</math>&nbsp;; as such the generators are [[involution (mathematics)|involution]]s.
* If <math>m_{ij} = 2</math>, then the generators <math>r_i</math> and <math>r_j</math> commute.  This follows by observing that
::<math>xx = yy = 1</math>,
: together with
::<math>xyxy = 1</math>
: implies that
::<math>xy = x(xyxy)y = (xx)yx(yy) = yx</math>.
:Alternatively, since the generators are involutions, <math>r_i = r_i^{-1}</math>, so <math>1 =(r_ir_j)^2=r_ir_jr_ir_j=r_ir_jr_i^{-1}r_j^{-1}</math>. That is to say, the [[commutator]] of <math>r_i</math> and <math>r_j</math> is equal to 1, or equivalently that <math>r_i</math> and <math>r_j</math> commute.

The reason that  <math>m_{ij} = m_{ji}</math> for <math>i \neq j</math> is stipulated in the definition is that
:<math>yy = 1</math>,
together with
:<math>(xy)^m = 1</math>
already implies that
:<math>(yx)^m = (yx)^myy = y(xy)^my = yy = 1</math>. 
An alternative proof of this implication is the observation that <math>(xy)^k</math> and <math>(yx)^k</math> are [[conjugate elements|conjugates]]: indeed <math>y(xy)^k y^{-1} = (yx)^k yy^{-1}=(yx)^k</math>.

===Coxeter matrix and Schläfli matrix===
The '''Coxeter matrix''' is the <math>n\times n</math> [[symmetric matrix]] with entries <math>m_{ij}</math>. Indeed, every symmetric matrix with diagonal entries exclusively 1 and nondiagonal entries in the set <math>\{2,3,\ldots\} \cup \{\infty\}</math> is a Coxeter matrix.

The Coxeter matrix can be conveniently encoded by a '''[[Coxeter–Dynkin diagram|Coxeter diagram]]''', as per the following rules.
* The vertices of the graph are labelled by generator subscripts.
* Vertices <math>i</math> and <math>j</math> are adjacent if and only if <math>m_{ij}\geq 3</math>.
* An edge is labelled with the value of <math>m_{ij}</math> whenever the value is <math>4</math> or greater.

In particular, two generators [[commutative operation|commute]] if and only if they are not joined by an edge. 
Furthermore, if a Coxeter graph has two or more [[connected component (graph theory)|connected component]]s, the associated group is the [[direct product of groups|direct product]] of the groups associated to the individual components.
Thus the [[disjoint union]] of Coxeter graphs yields a [[direct product of groups|direct product]] of Coxeter groups.

The Coxeter matrix, <math>M_{ij}</math>, is related to the <math>n\times n</math> [[Schläfli matrix]] <math>C</math> with entries <math>C_{ij} = -2\cos(\pi/M_{ij})</math>, but the elements are modified, being proportional to the [[dot product]] of the pairwise generators. The Schläfli matrix is useful because its [[eigenvalues]] determine whether the Coxeter group is of ''finite type'' (all positive), ''affine type'' (all non-negative, at least one zero), or ''indefinite type'' (otherwise). The indefinite type is sometimes further subdivided, e.g. into hyperbolic and other Coxeter groups. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups.

{| class=wikitable
|+ Examples
|- align=center
!Coxeter group
! A<sub>1</sub>×A<sub>1</sub>
! A<sub>2</sub>
! B<sub>2</sub>
! I<sub>2</sub>(5)
! G<sub>2</sub>
! <math>{\tilde{A}}_1 = I_2(\infty)</math>
! A<sub>3</sub>
! B<sub>3</sub>
! D<sub>4</sub>
! <math>{\tilde{A}}_3</math>
|- align=center
!Coxeter diagram
|{{CDD|node|2|node}}
|{{CDD|node|3|node}}
|{{CDD|node|4|node}}
|{{CDD|node|5|node}}
|{{CDD|node|6|node}}
|{{CDD|node|infin|node}}
|{{CDD|node|3|node|3|node}}
|{{CDD|node|4|node|3|node}}
|{{CDD|node|3|node|split1|nodes}}
|{{CDD|node|split1|nodes|split2|node}}
|- align=center
!Coxeter matrix
|<math>\left [
\begin{smallmatrix}
 1 & 2 \\
 2 & 1 \\ 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 3 \\
 3 & 1 \\ 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 4 \\
 4 & 1 \\ 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 5 \\
 5 & 1 \\ 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 6 \\
 6 & 1 \\ 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & \infty \\
 \infty & 1 \\ 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 3 & 2 \\
 3 & 1 & 3 \\
 2 & 3 & 1 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 4 & 2 \\
 4 & 1 & 3 \\
 2 & 3 & 1 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 3 & 2 & 2 \\
 3 & 1 & 3 & 3 \\
 2 & 3 & 1 & 2\\
 2 & 3 & 2 & 1 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 3 & 2 & 3 \\
 3 & 1 & 3 & 2 \\
 2 & 3 & 1 & 3\\
 3 & 2 & 3 & 1 
\end{smallmatrix}\right ]</math>
|- align=center
!Schläfli matrix
|<math>\left [
\begin{smallmatrix}
 2 & 0 \\
 0 & 2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 & -1 \\
 -1 & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 & -\sqrt2 \\
 -\sqrt2 & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 & -\phi \\
 -\phi & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 & -\sqrt3 \\
 -\sqrt3 & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 & -2 \\
 -2 & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 & -1  & \ \,0 \\
 -1  & \ \,2 & -1 \\
 \ \,0 & -1  & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,\ \ 2 & -\sqrt{2} & \ \,0 \\
 -\sqrt{2} & \ \,\ \ 2 &   -1 \\
 \ \,\ \ 0 &   \ \,-1 & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 &  -1 & \ \,0 & \ \,0 \\
  -1 & \ \,2 &  -1 &  -1 \\
 \ \,0 &  -1 & \ \,2 & \ \,0 \\
 \ \,0 &  -1 & \ \,0 & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 &  -1 & \ \,0 &  -1 \\
  -1 & \ \,2 &  -1 & \ \,0 \\
 \ \,0 &   -1 & \ \,2 &  -1 \\
  -1 & \ \,0 &  -1 & \ \,2 
\end{smallmatrix}\right ]</math>
|}

==An example==
The graph <math>A_n</math> in which [[Vertex (graph theory)|vertices]] <math>1</math> through <math>n</math> are placed in a row with each vertex joined by an unlabelled [[edge (graph theory)|edge]] to its immediate neighbors is the Coxeter diagram of the [[symmetric group]] <math>S_{n+1}</math>; the [[Generating set of a group|generators]] correspond to the [[Transposition (mathematics)|transpositions]] <math>(1~~2), (2~~3), \dots, (n~~n+1)</math>. Any two non-consecutive transpositions commute, while multiplying two consecutive transpositions gives a 3-cycle : <math>(k~~k+1) \cdot (k+1~~k+2) = (k~~k+2~~k+1)</math>. Therefore <math>S_{n+1}</math> is a [[quotient group|quotient]] of the Coxeter group having Coxeter diagram <math>A_n</math>. Further arguments show that this quotient map is an isomorphism.

==Abstraction of reflection groups==
{{Further|Reflection group}}
Coxeter groups are an abstraction of reflection groups. Coxeter groups are ''abstract'' groups, in the sense of being given via a presentation. On the other hand, reflection groups are ''concrete'', in the sense that each of its elements is the composite of finitely many  geometric reflections about linear hyperplanes in some euclidean space. Technically, a reflection group is a subgroup of a [[linear group]] (or various generalizations) generated by orthogonal matrices of determinant -1. Each generator of a Coxeter group has order 2, which abstracts the geometric fact that performing a reflection twice is the identity. Each relation of the form <math>(r_ir_j)^k</math>, corresponding to the geometric fact that, given two [[hyperplane]]s meeting at an angle of <math>\pi/k</math>, the composite of the two reflections about these hyperplanes is a rotation by <math>2\pi/k</math>, which has order ''k''.

In this way, every reflection group may be presented as a Coxeter group.<ref name="Coxeter1934"/> The converse is partially true: every finite Coxeter group admits a faithful [[linear representation|representation]] as a finite reflection group of some Euclidean space.<ref name="Coxeter1935"/>
However, not every infinite Coxeter group admits a representation as a reflection group.

Finite Coxeter groups have been classified.<ref name="Coxeter1935"/>

==Finite Coxeter groups==
{{Dark mode invert|[[File:Finite coxeter.svg|500px|right|thumb|Coxeter graphs of the irreducible finite Coxeter groups]]}}

===Classification===
Finite Coxeter groups are classified in terms of their [[Coxeter–Dynkin diagram|Coxeter diagrams]].<ref name="Coxeter1935"/>

The finite Coxeter groups with connected Coxeter diagrams consist of three one-parameter families of increasing dimension (<math>A_n</math> for <math>n \geq 1</math>, <math>B_n</math> for <math>n \geq 2</math>, and <math>D_n</math> for <math>n \geq 4</math>), a one-parameter family of dimension two (<math>I_2(p)</math> for <math>p \geq 5</math>), and six [[exceptional object|exceptional]] groups (<math>E_6, E_7, E_8, F_4, H_3,</math> and <math>H_4</math>). Every finite Coxeter group is the [[direct product]] of finitely many of these irreducible groups.{{efn|In some contexts, the naming scheme may be extended to allow the following alternative or redundant names: <math>B_1 \cong A_1</math>, <math>D_2 \cong I_2(2) \cong A_1 \times A_1</math>, <math>I_2(3) \cong A_2</math>, <math>I_2(4) \cong B_2</math>, <math>H_2 \cong I_2(5)</math>, and <math>D_3 \cong A_3</math>.}}  

===Weyl groups===
{{main|Weyl group}}
Many, but not all of these, are Weyl groups, and every Weyl group can be realized as a Coxeter group. The Weyl groups are the families <math>A_n, B_n,</math> and <math>D_n,</math> and the exceptions <math>E_6, E_7, E_8, F_4,</math> and <math>I_2(6),</math> denoted in Weyl group notation as <math>G_2.</math>

The non-Weyl ones are the exceptions <math>H_3</math> and <math>H_4,</math> and those members of the family <math>I_2(p)</math> that are not [[exceptional isomorphism|exceptionally isomorphic]] to a Weyl group (namely <math>I_2(3) \cong A_2, I_2(4) \cong B_2,</math> and <math>I_2(6) \cong G_2</math>).

This can be proven by comparing the restrictions on (undirected) [[Dynkin diagram]]s with the restrictions on Coxeter diagrams of finite groups: formally, the Coxeter graph can be obtained from the Dynkin diagram by discarding the direction of the edges, and replacing every double edge with an edge labelled 4 and every triple edge by an edge labelled 6. Also note that every finitely generated Coxeter group is an [[automatic group]].<ref name="BrinkAndHowlett">{{cite journal|last1=Brink|first1=Brigitte|last2=Howlett|first2=Robert B.|title=A finiteness property and an automatic structure for Coxeter groups|journal=Mathematische Annalen|volume=296|issue=1|pages=179–190|year=1993|doi=10.1007/BF01445101|zbl=0793.20036|s2cid=122177473}}</ref> Dynkin diagrams have the additional restriction that the only permitted edge labels are 2, 3, 4, and 6, which yields the above. Geometrically, this corresponds to the [[crystallographic restriction theorem]], and the fact that excluded polytopes do not fill space or tile the plane – for <math>H_3,</math> the dodecahedron (dually, icosahedron) does not fill space; for <math>H_4,</math> the 120-cell (dually, 600-cell) does not fill space; for <math>I_2(p)</math> a ''p''-gon does not tile the plane except for <math>p=3, 4,</math> or <math>6</math> (the triangular, square, and hexagonal tilings, respectively).

Note further that the (directed) Dynkin diagrams ''B<sub>n</sub>'' and ''C<sub>n</sub>'' give rise to the same Weyl group (hence Coxeter group), because they differ as ''directed'' graphs, but agree as ''undirected'' graphs – direction matters for root systems but not for the Weyl group; this corresponds to the [[hypercube]] and [[cross-polytope]] being different regular polytopes but having the same symmetry group.

===Properties===
Some properties of the finite irreducible Coxeter groups are given in the following table. The order of a reducible group can be computed by the product of its irreducible subgroup orders.
{{sort-under}}
{| class="wikitable sortable sort-under"
! {{verth|Rank ''n''}} || {{verth|Group<br />symbol}} || {{verth|Alternate<br />symbol}} || [[Coxeter notation|Bracket<br />notation]]||Coxeter<br />graph || data-sort-type="number"|Reflections<br />{{math|1=''m'' = {{sfrac|1|2}}''nh''}}<ref>{{cite book|title=Regular Polytopes|last=Coxeter|first=H. S. M.|chapter=12.6. The number of reflections|date=January 1973 |publisher=Courier Corporation |language=en|isbn=0-486-61480-8}}</ref>||data-sort-type="number"|[[Coxeter element|Coxeter number]]<br />''h''||data-sort-type="number"| [[Order (group theory)|Order]] || Group structure<ref name="wilson">{{Citation|last1=Wilson|first1=Robert A.|author-link=Robert Arnott Wilson|title=The finite simple groups|publisher=[[Springer-Verlag]]|location=Berlin, New York|series=[[Graduate Texts in Mathematics]] 251|isbn=978-1-84800-987-5|doi=10.1007/978-1-84800-988-2|year=2009|chapter=Chapter 2|volume=251}}</ref> || Related [[Uniform polytope|polytopes]]
|- align=center
!1 ||''A''<sub>1</sub> 
| ''A''<sub>1</sub> || [ ]|| {{CDD|node}} || 1 ||2 || 2 || <math>S_2</math> || { }
|- align=center
!2 ||''A''<sub>2</sub> 
| ''A''<sub>2</sub> || [3]|| {{CDD|node|3|node}} || 3 ||3 || 6 || <math>S_3\cong D_6\cong \operatorname{GO}^-_2(2)\cong \operatorname{GO}^+_2(4)</math> || [[equilateral triangle|{3}]] 
|- align=center
!3 ||''A''<sub>3</sub> 
| ''A''<sub>3</sub> || [3,3]|| {{CDD|node|3|node|3|node}} || 6 ||4 || 24 || <math>S_4</math> || [[regular tetrahedron|{3,3}]] 
|- align=center
!4 ||''A''<sub>4</sub> 
| ''A''<sub>4</sub> || [3,3,3]|| {{CDD|node|3|node|3|node|3|node}} || 10 ||5 || 120 || <math>S_5</math> || [[5-cell|{3,3,3}]] 
|- align=center
!5 ||''A''<sub>5</sub> 
| ''A''<sub>5</sub> || [3,3,3,3]|| {{CDD|node|3|node|3|node|3|node|3|node}} || 15 ||6 || 720 || <math>S_6</math> || [[5-simplex|{3,3,3,3}]] 
|- align=center
!''n'' ||''A''<sub>''n''</sub> || ''A''<sub>''n''</sub> || [3<sup>''n''−1</sup>]|| {{CDD|node|3|node|3}}...{{CDD|3|node|3|node}} || ''n''(''n'' + 1)/2 ||''n'' + 1 || (''n'' + 1)! || <math>S_{n+1}</math> || [[simplex|''n''-simplex]] 
|- align=center
!2 ||''B''<sub>2</sub>
| ''C''<sub>2</sub> || [4]|| {{CDD|node|4|node}} || 4 ||4 || 8 || <math>C_{2}\wr S_{2} \cong D_8\cong \operatorname{GO}^-_2(3)\cong \operatorname{GO}^+_2(5)</math> || [[square|{4}]]
|- align=center
!3 ||''B''<sub>3</sub>
| ''C''<sub>3</sub> || [4,3]|| {{CDD|node|4|node|3|node}}|| 9 ||6 || 48 || <math>C_{2}\wr S_{3}\cong S_4\times 2</math> || [[cube|{4,3}]] / [[regular octahedron|{3,4}]] 
|- align=center
!4 ||''B''<sub>4</sub>
| ''C''<sub>4</sub> || [4,3,3]|| {{CDD|node|4|node|3|node|3|node}}|| 16 ||8 || 384 || <math>C_{2}\wr S_{4}</math> || [[tesseract|{4,3,3}]] / [[16-cell|{3,3,4}]] 
|- align=center
!5 ||''B''<sub>5</sub>
| ''C''<sub>5</sub> || [4,3,3,3]|| {{CDD|node|4|node|3|node|3|node|3|node}} || 25 || 10 || 3840 || <math>C_{2}\wr S_{5}</math> || [[5-cube|{4,3,3,3}]] / [[5-orthoplex|{3,3,3,4}]] 
|- align=center
!''n'' ||''B''<sub>''n''</sub>|| ''C''<sub>''n''</sub> || [4,3<sup>''n''−2</sup>]|| {{CDD|node|4|node|3}}...{{CDD|3|node|3|node}}|| ''n''<sup>2</sup> ||2''n'' || 2<sup>''n''</sup> ''n''! || <math>C_{2}\wr S_{n}</math> || ''n''-cube / [[orthoplex|''n''-orthoplex]] 
|- align=center
!4 ||''D''<sub>4</sub> 
| ''B''<sub>4</sub> || [3<sup>1,1,1</sup>]|| {{CDD|nodes|split2|node|3|node}}|| 12 ||6 || 192 || <math>C_{2}^3 S_{4}\cong 2^{1+4}\colon S_3</math> || [[16-cell|h{4,3,3}]] / [[16-cell|{3,3<sup>1,1</sup>}]]
|- align=center
!5 ||''D''<sub>5</sub> 
| ''B''<sub>5</sub> || [3<sup>2,1,1</sup>]|| {{CDD|nodes|split2|node|3|node|3|node}} || 20 ||8 || 1920 || <math>C_{2}^4 S_{5}</math>|| [[5-demicube|h{4,3,3,3}]] / [[5-orthoplex|{3,3,3<sup>1,1</sup>}]]
|- align=center
!''n'' ||''D''<sub>''n''</sub> || ''B''<sub>''n''</sub> || [3<sup>''n''−3,1,1</sup>]|| {{CDD|nodes|split2|node|3}}...{{CDD|3|node|3|node}}|| ''n''(''n'' − 1) ||2(''n'' − 1) || 2<sup>''n''&minus;1</sup> ''n''! || <math>C_{2}^{n-1} S_{n}</math> || [[demihypercube|''n''-demicube]] / ''n''-orthoplex
|- align=center
!6 ||[[E6 (mathematics)|''E''<sub>6</sub>]]
|''E''<sub>6</sub> || [3<sup>2,2,1</sup>]|| {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}} || 36 ||12 || 51840 
|
<math>\operatorname{GO}_6^{-}(2) \cong \operatorname{SO}_5(3) \cong \operatorname{PSp}_4(3) \colon 2 \cong \operatorname{PSU}_4(2) \colon 2</math>
|
[[2 21 polytope|2<sub>21</sub>]], [[1 22 polytope|1<sub>22</sub>]] 
|- align=center
!7 ||[[E7 (mathematics)|''E''<sub>7</sub>]]
|''E''<sub>7</sub> || [3<sup>3,2,1</sup>]|| {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}|| 63 ||18 || 2903040 || <math> \operatorname{GO}_7(2)\times 2 \cong \operatorname{Sp}_6(2)\times 2 </math>|| [[3 21 polytope|3<sub>21</sub>]], [[2 31 polytope|2<sub>31</sub>]], [[1 32 polytope|1<sub>32</sub>]] 
|- align=center
!8 ||[[E8 (mathematics)|''E''<sub>8</sub>]]
| ''E''<sub>8</sub> || [3<sup>4,2,1</sup>]|| {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}|| 120 ||30 || 696729600 || <math>2\cdot\operatorname{GO}_8^{+}(2)</math>||[[4 21 polytope|4<sub>21</sub>]], [[2 41 polytope|2<sub>41</sub>]], [[1 42 polytope|1<sub>42</sub>]] 
|- align=center
!4 ||[[F4 (mathematics)|''F''<sub>4</sub>]]
|''F''<sub>4</sub> || [3,4,3]|| {{CDD|node|3|node|4|node|3|node}} || 24 ||12 || 1152 ||<math>\operatorname{GO}^+_4(3)\cong 2^{1+4}\colon(S_3 \times S_3)</math>|| [[24-cell|{3,4,3}]] 
|- align=center
!2 ||[[G2 (mathematics)|''G''<sub>2</sub>]]
|| – (''D''{{supsub|6|2}}) || [6]|| {{CDD|node|6|node}} || 6 ||6 || 12 || <math>D_{12}\cong \operatorname{GO}^-_2(5)\cong \operatorname{GO}^+_2(7)</math>|| [[hexagon|{6}]] 
|- align=center
!2 ||''I''<sub>2</sub>(5) 
|| ''G''<sub>2</sub> || [5]||{{CDD|node|5|node}} || 5 || 5 || 10 || <math>D_{10}\cong \operatorname{GO}^-_2(4)</math>|| [[pentagon|{5}]] 
|- align=center
!3 ||''H''<sub>3</sub> 
| ''G''<sub>3</sub> 
|| [3,5]|| {{CDD|node|5|node|3|node}} || 15 ||10 || 120 || <math>2\times A_5</math>|| [[icosahedron|{3,5}]] / [[dodecahedron|{5,3}]] 
|- align=center
!4 ||''H''<sub>4</sub> 
| ''G''<sub>4</sub> || [3,3,5]|| {{CDD|node|5|node|3|node|3|node}} || 60 ||30 || 14400 || <math>2\cdot(A_5\times A_5)\colon 2</math>{{efn|an index 2 subgroup of <math>\operatorname{GO}^+_4(5)</math>}}|| [[120-cell|{5,3,3}]] / [[600-cell|{3,3,5}]] 
|- align=center
!2 ||''I''<sub>2</sub>(''n'') 
|| ''D''{{supsub|''n''|2}}|| [''n'']|| {{CDD|node|n|node}} || ''n'' ||''n'' || 2''n'' 
|
<math>D_{2n}</math>

<math>\cong \operatorname{GO}^-_2(n-1)</math> when ''n'' = ''p''<sup>''k''</sup> + 1, ''p'' prime
<math>\cong \operatorname{GO}^+_2(n+1)</math> when ''n'' = ''p''<sup>''k''</sup> − 1, ''p'' prime

| [[regular polygon|{''p''}]] 
|}

===Symmetry groups of regular polytopes===
The symmetry group of every regular polytope is a finite Coxeter group. Note that [[dual polytope]]s have the same symmetry group.

There are three series of regular polytopes in all dimensions. The symmetry group of a regular ''n''-simplex is the symmetric group ''S''<sub>''n''+1</sub>, also known as the Coxeter group of type ''A<sub>n</sub>''. The symmetry group of the ''n''-[[cube]] and its dual, the ''n''-cross-polytope, is ''B<sub>n</sub>'', and is known as the [[hyperoctahedral group]].

The exceptional regular polytopes in dimensions two, three, and four, correspond to other Coxeter groups. In two dimensions, the [[dihedral group]]s, which are the symmetry groups of [[regular polygon]]s, form the series ''I''<sub>2</sub>(''p''), for ''p'' ≥ 3. In three dimensions, the symmetry group of the regular [[dodecahedron]] and its dual, the regular [[icosahedron]], is ''H''<sub>3</sub>, known as the [[full icosahedral group]]. In four dimensions, there are three exceptional regular polytopes, the [[24-cell]], the [[120-cell]], and the [[600-cell]]. The first has symmetry group ''F''<sub>4</sub>, while the other two are dual and have symmetry group ''H''<sub>4</sub>.

The Coxeter groups of type ''D''<sub>''n''</sub>, ''E''<sub>6</sub>, ''E''<sub>7</sub>, and ''E''<sub>8</sub> are the symmetry groups of certain [[semiregular polytope]]s.

==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.

==Hyperbolic Coxeter groups==
There are infinitely many [[Coxeter–Dynkin diagram#Hyperbolic Coxeter groups|hyperbolic Coxeter groups]] describing reflection groups in [[hyperbolic geometry|hyperbolic space]], notably including the hyperbolic triangle groups.

==Irreducible Coxeter groups==
A Coxeter group is said to be ''irreducible'' if its Coxeter–Dynkin diagram is connected.  Every Coxeter group is the [[direct product of groups|direct product]] of the irreducible groups that correspond to the [[Component (graph theory)|components]] of its Coxeter–Dynkin diagram.  

==Partial orders==
A choice of reflection generators gives rise to a [[length function]] ''ℓ'' on a Coxeter group, namely the minimum number of uses of generators required to express a group element; this is precisely the length in the [[word metric]] in the [[Cayley graph]]. An expression for ''v'' using ''ℓ''(''v'') generators is a ''reduced word''. For example, the permutation (13) in ''S''<sub>3</sub> has two reduced words, (12)(23)(12) and (23)(12)(23). The function <math>v \to (-1)^{\ell(v)}</math> defines a map <math>G \to \{\pm 1\},</math> generalizing the [[sign map]] for the symmetric group.

Using reduced words one may define three [[partial order]]s on the Coxeter group, the (right) '''[[weak Bruhat order|weak order]]''', the '''absolute order''' and the '''[[Bruhat order]]''' (named for [[François Bruhat]]). An element ''v'' exceeds an element ''u'' in the Bruhat order if some (or equivalently, any) reduced word for ''v'' contains a reduced word for ''u'' as a substring, where some letters (in any position) are dropped. In the weak order, ''v''&nbsp;≥&nbsp;''u'' if some reduced word for ''v'' contains a reduced word for ''u'' as an initial segment. Indeed, the word length makes this into a [[graded poset]]. The [[Hasse diagram]]s corresponding to these orders are objects of study, and are related to the [[Cayley graph]] determined by the generators.  The absolute order is defined analogously to the weak order, but with generating set/alphabet consisting of all conjugates of the Coxeter generators.

For example, the permutation (1 2 3) in ''S''<sub>3</sub> has only one reduced word, (12)(23), so covers (12) and (23) in the Bruhat order but only covers (12) in the weak order.

==Homology==
Since a Coxeter group <math>W</math> is generated by finitely many elements of order 2, its [[abelianization]] is an [[elementary abelian group|elementary abelian 2-group]], i.e., it is isomorphic to the direct sum of several copies of the [[cyclic group]] <math>Z_2</math>. This may be restated in terms of the first [[Group cohomology#Group homology|homology group]] of <math>W</math>.

The [[Schur multiplier]] <math>M(W)</math>, equal to the second homology group of <math>W</math>, was computed in {{Harv|Ihara|Yokonuma|1965}} for finite reflection groups and in {{Harv|Yokonuma|1965}} for affine reflection groups, with a more unified account given in {{Harv|Howlett|1988}}. In all cases, the Schur multiplier is also an elementary abelian 2-group. For each infinite family <math>\{W_n\}</math> of finite or affine Weyl groups, the rank of <math>M(W_n)</math> stabilizes as <math>n</math> goes to infinity.

==See also==
* [[Artin–Tits group]]
* [[Chevalley–Shephard–Todd theorem]]
* [[Complex reflection group]]
* [[Coxeter element]]
* [[Iwahori–Hecke algebra]], a quantum deformation of the [[group ring|group algebra]]
* [[Kazhdan–Lusztig polynomial]]
* [[Longest element of a Coxeter group]]
* [[Parabolic subgroup of a reflection group]]
* [[Supersolvable arrangement]]
* [[Isomorphism problem of Coxeter groups]]

==Notes==
{{notelist}}

==References==
{{Reflist}}

==Bibliography==
{{refbegin}}
*{{cite book|last=Hall|first=Brian C.|title=Lie groups, Lie algebras, and representations: An elementary introduction|edition=2nd|series=Graduate Texts in Mathematics|volume=222|publisher=Springer|year=2015|isbn=978-3-319-13466-6}}
*{{cite journal|first1=S.|last1=Ihara|first2=Takeo|last2=Yokonuma|title=On the second cohomology groups (Schur-multipliers) of finite reflection groups|year=1965|journal=J. Fac. Sci. Univ. Tokyo, Sect. 1|volume=11|pages=155–171|zbl=0136.28802|url=http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6049/1/jfs110203.pdf|archive-url=https://web.archive.org/web/20131023064852/http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6049/1/jfs110203.pdf|archive-date=2013-10-23}}
*{{cite journal|title=On the Schur Multipliers of Coxeter Groups|first=Robert B.|last=Howlett |journal=J. London Math. Soc.|year=1988|series=2|volume=38|issue=2|pages=263–276|doi=10.1112/jlms/s2-38.2.263|zbl=0627.20019}}
*{{cite journal|first=Takeo|last=Yokonuma|title=On the second cohomology groups (Schur-multipliers) of infinite discrete reflection groups|year=1965|journal=J. Fac. Sci. Univ. Tokyo, Sect. 1|volume=11|pages=173–186|zbl=0136.28803|hdl=2261/6049}}
{{refend}}

==Further reading==
{{refbegin}}
*{{cite book|first1=Anders|last1=Björner|first2=Francesco|last2=Brenti|title=Combinatorics of Coxeter Groups|url=https://books.google.com/books?id=1TBPz5sd8m0C|year=2005|publisher=Springer|isbn=978-3-540-27596-1|series=[[Graduate Texts in Mathematics]]|volume=231|author1-link=Anders Björner|zbl=1110.05001}}
*{{cite book|first2=Clark T.|last2=Benson|first1=Larry C.|last1=Grove|title=Finite Reflection Groups|url=https://books.google.com/books?id=525Gh4uzjnIC|year=1985|publisher=Springer|series=Graduate texts in mathematics|volume=99|isbn=978-0-387-96082-1}}
*{{cite book|first=Richard|last=Kane|title=Reflection Groups and Invariant Theory|url=https://books.google.com/books?id=KmL1uuiMyFUC&pg=PP10|year=2001|publisher=Springer|series=CMS Books in Mathematics|isbn=978-0-387-98979-2|zbl=0986.20038}}
*{{cite book|first=Howard|last=Hiller|title=Geometry of Coxeter groups|url=https://books.google.com/books?id=7jzvAAAAMAAJ|year=1982|publisher=Pitman|isbn=978-0-273-08517-1|series=Research Notes in Mathematics|volume=54|zbl=0483.57002}}
*{{cite journal|first=Ernest B.|last=Vinberg|author-link=Ernest Vinberg|title=Absence of crystallographic groups of reflections in Lobachevski spaces of large dimension|journal=Trudy Moskov. Mat. Obshch.|volume=47|year=1984}}
{{refend}}

==External links==
* {{springer|title=Coxeter group|id=p/c026980}}
* {{MathWorld|urlname=CoxeterGroup|title=Coxeter group|mode=cs2}}
* {{Citation|url=http://www.jenn3d.org/index.html|title= Jenn software for visualizing the Cayley graphs of finite Coxeter groups on up to four generators}}

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{{DEFAULTSORT:Coxeter Group}}
[[Category:Coxeter groups| ]]
[[Category:Reflection groups|*]]