Editing Coxeter group (section)

==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.