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| In [[mathematics]], a '''Coxeter group''', named after [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]], is an [[group (mathematics)|abstract group]] that admits a [[group presentation|formal description]] in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite [[reflection group|Euclidean reflection group]]s; the [[symmetry group]]s of [[regular polyhedron|regular polyhedra]] are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced {{Harv|Coxeter|1934}} as abstractions of [[reflection group]]s, and finite Coxeter groups were classified in 1935 {{Harv|Coxeter|1935}}.
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| Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the [[symmetry group]]s 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#Regular and irregular tessellations|regular tessellation]]s of the [[Euclidean plane]] and the [[Hyperbolic space|hyperbolic plane]], and the Weyl groups of infinite-dimensional [[Kac–Moody algebra]]s.
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| Standard references include {{Harv|Humphreys|1990}} and {{Harv|Davis|2007}}.
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| ==Definition==
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| Formally, a '''Coxeter group''' can be defined as a [[group (mathematics)|group]] with the [[presentation of a group|presentation]]
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| :<math>\left\langle r_1,r_2,\ldots,r_n \mid (r_ir_j)^{m_{ij}}=1\right\rangle</math>
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| where <math>m_{ii}=1</math> and <math>m_{ij}\geq 2</math> for <math>i\neq j</math>.
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| The condition <math>m_{i j}= \infty</math> means no relation of the form <math>(r_i r_j)^m</math> should be imposed.
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| The pair ''(W,S)'' where ''W'' is a Coxeter group with generators ''S''={''r''<sub>''1''</sub>,...,''r''<sub>''n''</sub>} is called '''Coxeter system'''. Note that in general ''S'' is ''not'' uniquely determined by ''W''. For example, the Coxeter groups of type ''BC''<sub>''3''</sub> and ''A''<sub>''1''</sub>x''A''<sub>''3''</sub> are isomorphic but the Coxeter systems are not equivalent (see below for an explanation of this notation).
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| A number of conclusions can be drawn immediately from the above definition.
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| * The relation ''m''<sub>''i i''</sub> = 1 means that (''r''<sub>''i''</sub>''r''<sub>''i''</sub> )<sup>1</sup> = (''r''<sub>''i''</sub> )<sup>2</sup> = 1 for all ''i'' ; the generators are [[involution (mathematics)|involution]]s.
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| * If ''m''<sub>''i j''</sub> = 2, then the generators ''r''<sub>''i''</sub> and ''r''<sub>''j''</sub> commute. This follows by observing that
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| ::''xx'' = ''yy'' = 1,
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| : together with
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| :: ''xyxy'' = 1
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| : implies that
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| :: ''xy'' = ''x''(''xyxy'')''y'' = (''xx'')''yx''(''yy'') = ''yx''.
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| :Alternatively, since the generators are involutions, <math>r_i = r_i^{-1}</math>, so <math>(r_ir_j)^2=r_ir_jr_ir_j=r_ir_jr_i^{-1}r_j^{-1}</math>, and thus is equal to the [[commutator]].
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| * In order to avoid redundancy among the relations, it is necessary to assume that ''m<sub>i j</sub>'' = ''m<sub>j i</sub>''. This follows by observing that
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| ::''yy'' = 1,
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| : together with
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| :: (''xy'')<sup>''m''</sup> = 1
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| : implies that
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| :: (''yx'')<sup>''m''</sup> = (''yx'')<sup>''m''</sup>''yy'' = ''y''(''xy'')<sup>''m''</sup>''y'' = ''yy'' = 1.
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| :Alternatively, <math>(xy)^m</math> and <math>(yx)^m</math> are [[conjugate elements]], as <math>y(xy)^m y^{-1} = (yx)^m yy^{-1}=(yx)^m</math>.
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| === Coxeter matrix and Schläfli matrix ===
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| The '''Coxeter matrix''' is the ''n''×''n'', [[symmetric matrix]] with entries ''m<sub>i j</sub>''. Indeed, every symmetric matrix with positive integer and ∞ entries and with 1's on the diagonal such that all nondiagonal entries are greater than 1 serves to define a Coxeter group.
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| The Coxeter matrix can be conveniently encoded by a '''[[Coxeter-Dynkin diagram|Coxeter diagram]]''', as per the following rules.
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| * The vertices of the graph are labelled by generator subscripts.
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| * Vertices ''i'' and ''j'' are connected if and only if ''m''<sub>''i j''</sub> ≥ 3.
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| * An edge is labelled with the value of ''m''<sub>''i j''</sub> whenever it is 4 or greater.
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| In particular, two generators [[commutative operation|commute]] if and only if they are not connected by an edge.
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| 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.
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| Thus the [[disjoint union]] of Coxeter graphs yields a [[direct product of groups|direct product]] of Coxeter groups.
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| The Coxeter matrix, M<sub>i,j</sub>, is related to the [[Schläfli matrix]], C<sub>i,j</sub>, but the elements are modified, being proportional to the [[dot product]] of the pairwise generators: Schläfli matrix C<sub>i,j</sub>=-2cos(π/M<sub>i,j</sub>). 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.
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| {| class=wikitable
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| |+ Examples
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| |- align=center
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| !Coxeter group
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| ! A<sub>1</sub>×A<sub>1</sub>
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| ! A<sub>2</sub>
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| ! <math>{\tilde{I}}_1</math>
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| ! A<sub>3</sub>
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| ! BC<sub>3</sub>
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| ! D<sub>4</sub>
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| ! <math>{\tilde{A}}_3</math>
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| |- align=center
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| ![[Coxeter diagram]]
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| |{{CDD|node|2|node}}
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| |{{CDD|node|3|node}}
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| |{{CDD|node|infin|node}}
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| |{{CDD|node|3|node|3|node}}
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| |{{CDD|node|4|node|3|node}}
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| |{{CDD|node|3|node|split1|nodes}}
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| |{{CDD|node|split1|nodes|split2|node}}
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| |- align=center
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| !Coxeter matrix
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| |<math>\left [
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| \begin{smallmatrix}
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| 1 & 2 \\
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| 2 & 1 \\
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| \end{smallmatrix}\right ]</math>
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| |<math>\left [
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| \begin{smallmatrix}
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| 1 & 3 \\
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| 3 & 1 \\
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| \end{smallmatrix}\right ]</math>
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| |<math>\left [
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| \begin{smallmatrix}
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| 1 & \infty \\
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| \infty & 1 \\
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| \end{smallmatrix}\right ]</math>
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| |<math>\left [
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| \begin{smallmatrix}
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| 1 & 3 & 2 \\
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| 3 & 1 & 3 \\
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| 2 & 3 & 1
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| \end{smallmatrix}\right ]</math>
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| |<math>\left [
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| \begin{smallmatrix}
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| 1 & 4 & 2 \\
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| 4 & 1 & 3 \\
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| 2 & 3 & 1
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| \end{smallmatrix}\right ]</math>
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| |<math>\left [
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| \begin{smallmatrix}
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| 1 & 3 & 2 & 2 \\
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| 3 & 1 & 3 & 3 \\
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| 2 & 3 & 1 & 2\\
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| 2 & 3 & 2 & 1
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| \end{smallmatrix}\right ]</math>
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| |<math>\left [
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| \begin{smallmatrix}
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| 1 & 3 & 2 & 3 \\
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| 3 & 1 & 3 & 2 \\
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| 2 & 3 & 1 & 3\\
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| 3 & 2 & 3 & 1
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| \end{smallmatrix}\right ]</math>
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| |- align=center
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| ![[Schläfli matrix]]
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| |<math>\left [
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| \begin{smallmatrix}
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| 2 & 0 \\
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| 0 & 2
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| \end{smallmatrix}\right ]</math>
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| |<math>\left [
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| \begin{smallmatrix}
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| 2 & -1 \\
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| -1 & 2
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| \end{smallmatrix}\right ]</math>
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| |<math>\left [
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| \begin{smallmatrix}
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| 2 & -2 \\
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| -2 & 2
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| \end{smallmatrix}\right ]</math>
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| |<math>\left [
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| \begin{smallmatrix}
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| 2 & -1 & 0 \\
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| -1 & 2 & -1 \\
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| 0 & -1 & 2
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| \end{smallmatrix}\right ]</math>
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| |<math>\left [
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| \begin{smallmatrix}
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| 2 & -\sqrt{2} & 0 \\
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| -\sqrt{2} & 2 & -1 \\
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| 0 & -1 & 2
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| \end{smallmatrix}\right ]</math>
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| |<math>\left [
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| \begin{smallmatrix}
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| 2 & -1 & 0 & 0 \\
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| -1 & 2 & -1 & -1 \\
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| 0 & -1 & 2 & 0\\
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| 0 & -1 & 0 & 2
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| \end{smallmatrix}\right ]</math>
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| |<math>\left [
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| \begin{smallmatrix}
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| 2 & -1 & 0 & -1 \\
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| -1 & 2 & -1 & 0 \\
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| 0 & -1 & 2 & -1\\
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| -1 & 0 & -1 & 2
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| \end{smallmatrix}\right ]</math>
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| |}
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| == An example ==
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| The graph in which [[Vertex (graph theory)|vertices]] 1 through ''n'' are placed in a row with each vertex connected by an unlabelled [[edge (graph theory)|edge]] to its immediate neighbors gives rise to the [[symmetric group]] ''S''<sub>''n''+1</sub>; the [[Generating set of a group|generators]] correspond to the [[Transposition (mathematics)|transpositions]] (1 2), (2 3), ... (''n'' ''n''+1). Two non-consecutive transpositions always commute, while (''k'' ''k''+1) (''k''+1 ''k''+2) gives the 3-cycle (''k'' ''k''+2 ''k''+1). Of course this only shows that ''S<sub>n+1</sub>'' is a [[quotient group]] of the Coxeter group described by the graph, but it is not too difficult to check that equality holds.
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| == Connection with reflection groups ==
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| {{details|Reflection group}}
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| Coxeter groups are deeply connected with [[reflection group]]s. Simply put, Coxeter groups are ''abstract'' groups (given via a presentation), while reflection groups are ''concrete'' groups (given as subgroups of [[linear group]]s or various generalizations). Coxeter groups grew out of the study of reflection groups — they are an abstraction: a reflection group is a subgroup of a linear group generated by reflections (which have order 2), while a Coxeter group is an abstract group generated by involutions (elements of order 2, abstracting from reflections), and whose relations have a certain form (<math>(r_ir_j)^k</math>, corresponding to hyperplanes meeting at an angle of <math>\pi/k</math>, with <math>r_ir_j</math> being of order ''k'' abstracting from a rotation by <math>2\pi/k</math>).
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| The abstract group of a reflection group is a Coxeter group, while conversely a reflection group can be seen as a [[linear representation]] of a Coxeter group. For ''finite'' reflection groups, this yields an exact correspondence: every finite Coxeter group admits a faithful representation as a finite reflection group of some Euclidean space. For infinite Coxeter groups, however, a Coxeter group may not admit a representation as a reflection group.
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| Historically, {{Harv|Coxeter|1934}} proved that every reflection group is a Coxeter group (i.e., has a presentation where all relations are of the form <math>r_i^2</math> or <math>(r_ir_j)^k</math>), and indeed this paper introduced the notion of a Coxeter group, while {{Harv|Coxeter|1935}} proved that every finite Coxeter group had a representation as a reflection group, and classified finite Coxeter groups.
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| == Finite Coxeter groups ==
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| [[Image:Finite coxeter.svg|500px|right|thumb|Coxeter graphs of the finite Coxeter groups.]]
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| === Classification ===
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| The finite Coxeter groups were classified in {{Harv|Coxeter|1935}}, in terms of [[Coxeter–Dynkin diagram]]s; they are all represented by [[reflection group]]s of finite-dimensional Euclidean spaces.
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| The finite Coxeter groups consist of three one-parameter families of increasing rank <math>A_n, BC_n, D_n,</math> one one-parameter family of dimension two, <math>I_2(p),</math> and six [[exceptional object|exceptional]] groups: <math>E_6, E_7, E_8, F_4, H_3,</math> and <math>H_4.</math>
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| ===Weyl groups===
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| {{main|Weyl group}}
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| 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, BC_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 groups are the exceptions <math>H_3</math> and <math>H_4,</math> and the family <math>I_2(p)</math> except where this coincides with one of the Weyl groups (namely <math>I_2(3) \cong A_2, I_2(4) \cong BC_2,</math> and <math>I_2(6) \cong G_2</math>).
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| 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–Dynkin diagram|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">{{Citation | last = Brink and Howlett | title = A finiteness property and an automatic structure for Coxeter groups | year = 1993 | journal = Mathematische Annalen | publisher = Springer Berlin / Heidelberg | issn= 0025-5831 | postscript = .}}</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).
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| 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.
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| ===Properties===
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| Some properties of the finite Coxeter groups are given in the following table:
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| {| class="wikitable"
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| !Group<BR>symbol || Alternate<BR>symbol || [[Coxeter notation|Bracket notation]] || Rank || [[Order (group theory)|Order]] || Related [[Uniform polytope|polytopes]] || [[Coxeter-Dynkin diagram]]
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| |- align=center
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| !''A''<sub>''n''</sub>
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| || ''A''<sub>''n''</sub> || [3<sup>n-1</sup>] || ''n'' || (''n'' + 1)! || ''n''-[[simplex]] || {{CDD|node|3|node|3}}..{{CDD|3|node|3|node}}
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| |- align=center
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| !''BC''<sub>''n''</sub>
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| || ''C''<sub>''n''</sub> || [4,3<sup>n-2</sup>]|| ''n'' || 2<sup>''n''</sup> ''n''! || ''n''-[[hypercube]] / ''n''-[[cross-polytope]] || {{CDD|node|4|node|3}}...{{CDD|3|node|3|node}}
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| |- align=center
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| !''D''<sub>''n''</sub>
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| || ''B''<sub>''n''</sub> || [3<sup>n-3,1,1</sup>]|| ''n'' || 2<sup>''n''−1</sup> ''n''! || ''n''-[[demihypercube]] || {{CDD|nodes|split2|node|3}}...{{CDD|3|node|3|node}}
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| |- align=center
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| ![[E6 (mathematics)|''E''<sub>6</sub>]]
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| || ''E''<sub>6</sub> || [3<sup>2,2,1</sup>] || 6 || 72x6! = 51840 || [[2 21 polytope|2<sub>21</sub>]], [[1 22 polytope|1<sub>22</sub>]] || {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}} or {{CDD|node|3|node|split1|nodes|3ab|nodes}}
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| |- align=center
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| ![[E7 (mathematics)|''E''<sub>7</sub>]]
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| ||''E''<sub>7</sub> || [3<sup>3,2,1</sup>]|| 7 || 72x8! = 2903040 || [[3 21 polytope|3<sub>21</sub>]], [[2 31 polytope|2<sub>31</sub>]], [[1 32 polytope|1<sub>32</sub>]] || {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}
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| |- align=center
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| ![[E8 (mathematics)|''E''<sub>8</sub>]]
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| || ''E''<sub>8</sub> || [3<sup>4,2,1</sup>]|| 8 || 192x10! = 696729600 || [[4 21 polytope|4<sub>21</sub>]], [[2 41 polytope|2<sub>41</sub>]], [[1 42 polytope|1<sub>42</sub>]] || {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
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| |- align=center
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| ![[F4 (mathematics)|''F''<sub>4</sub>]]
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| ||''F''<sub>4</sub> || [3,4,3] || 4 || 1152 || [[24-cell]] || {{CDD|node|3|node|4|node|3|node}}
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| |- align=center
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| ![[G2 (mathematics)|''G''<sub>2</sub>]]
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| || - || [6] || 2 || 12 || [[hexagon]] || {{CDD|node|6|node}}
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| |- align=center
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| !''H''<sub>2</sub>
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| || ''G''<sub>2</sub> || [5] || 2 || 10 || [[pentagon]] || {{CDD|node|5|node}}
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| |- align=center
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| !''H''<sub>3</sub>
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| || ''G''<sub>3</sub> || [3,5] || 3 || 120 || [[icosahedron]] / [[dodecahedron]] || {{CDD|node|5|node|3|node}}
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| |- align=center
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| !''H''<sub>4</sub>
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| || ''G''<sub>4</sub> || [3,3,5] || 4 || 14400 || [[120-cell]] / [[600-cell]] || {{CDD|node|5|node|3|node|3|node}}
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| |- align=center
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| !''I''<sub>2</sub>(''p'')
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| || ''D''<sub>2</sub><sup>''p''</sup> || [p] || 2 || 2''p'' || [[regular polygon|''p''-gon]] || {{CDD|node|p|node}}
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| |}
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| ===Symmetry groups of regular polytopes===
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| All [[symmetry group]]s of [[regular polytope]]s are finite Coxeter groups. Note that [[dual polytope]]s have the same symmetry group.
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| 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 ''BC<sub>n</sub>,'' and is known as the [[hyperoctahedral group]]. | |
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| 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''). 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 special 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>.
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| 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 [[Thorold Gosset|semiregular polytopes]].
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| {{:Polytope families}}
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| == Affine Coxeter groups ==
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| [[Image:Affine coxeter.PNG|320px|thumb|Coxeter diagrams for the Affine Coxeter groups]]
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| {{see also|Affine Dynkin diagram|Affine root system}}
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| 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 is obtained from the Coxeter graph of the Coxeter group by adding an additional vertex and one or two additional edges. For example, for ''n'' ≥ 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>''. For ''n'' = 2, this can be pictured as the symmetry group of the standard tiling of the plane by equilateral triangles.
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| A list of the affine Coxeter groups follows:
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| {| class="wikitable"
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| !Group<BR>symbol || [[Ernst Witt|Witt]]<BR>symbol || [[Coxeter notation|Bracket notation]] || Related uniform tessellation(s) || [[Coxeter-Dynkin diagram]]
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| |- align=center
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| !<math>{\tilde{A}}_n</math>
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| ||''P''<sub>''n+1''</sub> || [3<sup>[n]</sup>] || [[Simplectic honeycomb]] || {{CDD|node|split1|nodes|3ab}}...{{CDD|3ab|nodes|3ab|branch}}<BR>or<BR>{{CDD|branch|3ab|nodes|3ab}}...{{CDD|3ab|nodes|3ab|branch}}
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| |- align=center
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| !<math>{\tilde{B}}_n</math>
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| ||''S''<sub>''n+1''</sub> || [4,3<sup>n-3</sup>,3<sup>1,1</sup>] || [[Demihypercubic honeycomb]] || {{CDD|node|4|node|3|node|3}}...{{CDD|3|node|split1|nodes}}
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| |- align=center
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| !<math>{\tilde{C}}_n</math>
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| ||''R''<sub>''n+1''</sub> || [4,3<sup>n-2</sup>,4] || [[Hypercubic honeycomb]] || {{CDD|node|4|node|3|node|3}}...{{CDD|3|node|4|node}}
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| |- align=center
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| !<math>{\tilde{D}}_n</math>
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| ||''Q''<sub>''n+1''</sub> || [ 3<sup>1,1</sup>,3<sup>n-4</sup>,3<sup>1,1</sup>] ||[[Demihypercubic honeycomb]] || {{CDD|nodes|split2|node|3|node|3}}...{{CDD|3|node|split1|nodes}}
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| |- align=center
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| !<math>{\tilde{E}}_6</math>
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| ||''T''<sub>''7''</sub> || [3<sup>2,2,2</sup>] || [[2 22 honeycomb|2<sub>22</sub>]] || {{CDD|nodea|3a|nodea|3a|branch|3ab|nodes|3a|nodea}} or {{CDD|nodes|3ab|nodes|split2|node|3|node|3|node}}
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| |- align=center
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| !<math>{\tilde{E}}_7</math>
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| ||''T''<sub>''8''</sub> || [3<sup>3,3,1</sup>] || [[3 31 honeycomb|3<sub>31</sub>]], [[1 33 honeycomb|1<sub>33</sub>]] || {{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}}
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| |- align=center
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| !<math>{\tilde{E}}_8</math>
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| ||''T''<sub>''9''</sub> || [3<sup>5,2,1</sup>] || [[5 21 honeycomb|5<sub>21</sub>]], [[2 51 honeycomb|2<sub>51</sub>]], [[1 52 honeycomb|1<sub>52</sub>]] || {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
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| |- align=center
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| !<math>{\tilde{F}}_4</math>
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| ||''U''<sub>''5''</sub> || [3,4,3,3] || [[16-cell honeycomb]]<BR>[[24-cell honeycomb]]|| {{CDD|node|3|node|4|node|3|node|3|node}}
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| |- align=center
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| !<math>{\tilde{G}}_2</math>
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| ||''V''<sub>''3''</sub> || [6,3] || [[Hexagonal tiling]] and<BR>[[Triangular tiling]] || {{CDD|node|6|node|3|node}}
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| |- align=center
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| ! <math>{\tilde{I}}_1</math>
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| ||''W''<sub>''2''</sub> || [∞] || [[apeirogon]] || {{CDD|node|infin|node}}
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| |}
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| The 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.
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| ==Hyperbolic Coxeter groups==
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| 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 group]]s.
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| ==Partial orders==
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| 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)^{l(v)}</math> defines a map <math>G \to \{\pm 1\},</math> generalizing the [[sign map]] for the symmetric group.
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| 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 ≥ 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.
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| 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.
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| == Homology ==
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| Since a Coxeter group ''W'' 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]] '''Z'''<sub>2</sub>. This may be restated in terms of the first [[group homology|homology group]] of ''W''.
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| The [[Schur multiplier]] ''M''(''W'') (related to the second homology) 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 {''W''<sub>''n''</sub>} of finite or affine Weyl groups, the rank of ''M''(''W'') stabilizes as ''n'' goes to infinity.
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| ==See also==
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| * [[Artin group]]
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| * [[Triangle group]]
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| * [[Coxeter element]]
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| * [[Coxeter number]]
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| * [[Complex reflection group]]
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| * [[Chevalley–Shephard–Todd theorem]]
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| * [[Hecke algebra]], a quantum deformation of the [[group algebra]]
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| * [[Kazhdan–Lusztig polynomial]]
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| * [[Longest element of a Coxeter group]]
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| * [[Supersoluble arrangement]]
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| {{Reflist}}
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| ==Further reading==
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| * {{Citation
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| |doi=10.2307/1968753
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| |authorlink=H.S.M. Coxeter
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| |first=H.S.M.
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| |last=Coxeter
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| |title=Discrete groups generated by reflections
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| |journal=Ann. Of Math.
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| |volume=35
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| |year=1934
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| |issue=3
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| |pages=588–621
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| |jstor=1968753
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| }}
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| * {{Citation
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| |authorlink=H.S.M. Coxeter
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| |first=H.S.M.
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| |last=Coxeter
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| |title=The complete enumeration of finite groups of the form <math>r_i^2=(r_ir_j)^{k_{ij}}=1</math>
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| |journal=J. London Math. Soc.
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| |volume=10
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| |year=1935
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| |pages=21–25
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| }}
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| * {{citation | title = The Geometry and Topology of Coxeter Groups
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| |first = Michael W. | last = Davis | year = 2007 | url = http://www.math.osu.edu/~mdavis/davisbook.pdf | isbn = 978-0-691-13138-2 }}
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| * Larry C Grove and Clark T. Benson, ''Finite Reflection Groups'', Graduate texts in mathematics, vol. 99, Springer, (1985)
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| * James E. Humphreys, ''Reflection Groups and Coxeter Groups'', Cambridge studies in advanced mathematics, 29 (1990)
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| * Richard Kane, ''Reflection Groups and Invariant Theory'', CMS Books in Mathematics, Springer (2001)
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| * [[Anders Björner]] and Francesco Brenti, ''Combinatorics of Coxeter Groups'', [[Graduate Texts in Mathematics]], vol. 231, Springer, (2005)
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| *Howard Hiller, ''Geometry of Coxeter groups.'' Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. iv+213 pp. ISBN 0-273-08517-4
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| *Nicolas Bourbaki, ''Lie Groups and Lie Algebras: Chapter 4-6'', Elements of Mathematics, Springer (2002). ISBN 978-3-540-42650-9
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| | |
| *{{Citation
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| |title=On the Schur Multipliers of Coxeter Groups
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| |first=Robert B.
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| |last=Howlett
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| |journal=Journal of the London Mathematical Society
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| |year=1988
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| |series = 2
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| |volume=38
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| |issue=2
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| |pages=263–276
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| |doi=10.1112/jlms/s2-38.2.263
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| }}
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| * {{citation| first =E. B. |last=Vinberg| author-link=E. B. Vinberg|title=Absence of crystallographic groups of reflections in Lobachevski spaces of large dimension|journal=Trudy Moskov. Mat. Obshch. |volume=47|year=1984}}
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| | |
| * {{Citation
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| |first1 = S.
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| |last1 = Ihara
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| |first2 = Takeo
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| |last2 = Yokonuma
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| |title = On the second cohomology groups (Schur-multipliers) of finite reflection groups
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| |year = 1965
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| |journal = Jour. Fac. Sci. Univ. Tokyo, Sect. 1
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| |volume = 11
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| |pages = 155–171
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| |url = http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6049/1/jfs110203.pdf
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| }}
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| * {{Citation
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| |first = Takeo
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| |last = Yokonuma
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| |title = On the second cohomology groups (Schur-multipliers) of infinite discrete reflection groups
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| |year = 1965
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| |journal = Jour. Fac. Sci. Univ. Tokyo, Sect. 1
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| |volume = 11
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| |pages = 173–186
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| }}
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| {{refend}}
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| ==External links==
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| * {{springer|title=Coxeter group|id=p/c026980}}
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| * {{MathWorld | urlname=CoxeterGroup | title=Coxeter group }}
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| * {{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}}
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| [[Category:Coxeter groups| ]]
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| [[sl:Coxeterjeva grupa]]
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