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{{For|the Reflection Group On  the Future of Europe|European Council}}
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In [[group theory]] and [[geometry]], a '''reflection group''' is a [[discrete group]] which is generated by a set of [[reflection (mathematics)|reflections]] of a finite-dimensional [[Euclidean space]]. The symmetry group of a [[regular polytope]] or of a [[tessellation|tiling]] of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include [[Weyl group]]s and crystallographic [[Coxeter group]]s. While the [[orthogonal group]] is generated by reflections (by the [[Cartan–Dieudonné theorem]]), it is a continuous group (indeed, [[Lie group]]), not a discrete group, and is generally considered separately.
 
== Definition ==
 
Let ''E'' be a finite-dimensional [[Euclidean space]].  A '''finite reflection group''' is a subgroup of the [[general linear group]] of ''E'' which is generated by a set of orthogonal [[reflection (mathematics)|reflection]]s across hyperplanes passing through the origin. An '''affine reflection group''' is a discrete subgroup of the [[affine group]] of ''E'' that is generated by a set of ''affine reflections'' of ''E'' (without the requirement that the reflection hyperplanes pass through the origin).
 
The corresponding notions can be defined over other [[field (mathematics)|fields]], leading to '''[[complex reflection group]]s''' and analogues of reflection groups over a [[finite field]].
 
== Examples ==
=== Plane ===
 
In two dimensions, the finite reflection groups are the [[dihedral group]]s, which are generated by reflection in two lines that form an angle of <math>2\pi/n</math> and correspond to the [[Coxeter diagram]] <math>I_2(n).</math> Conversely, the cyclic [[point groups in two dimensions]] are ''not'' generated by reflections, and indeed contain no reflections – they are however subgroups of index 2 of a dihedral group.
 
Infinite reflection groups include the [[frieze group]]s <math>*\infty\infty</math> and <math>*22\infty</math> and the [[wallpaper group]]s <math>**</math>, <math>*2222</math>,<math>*333</math>,<math>*442</math> and <math>*632</math>. If the angle between two lines is an irrational multiple of pi, the group generated by reflections in these lines is infinite and non-discrete, hence, it is not a reflection group.
 
=== Space ===
 
Finite reflection groups are the [[point groups in three dimensions|point groups]] ''C<sub>nv</sub>'', ''D<sub>nh</sub>'', and the [[symmetry group]]s of the five [[Platonic solid]]s. Dual regular polyhedra (cube and octahedron, as well as dodecahedron and icosahedron) give rise to isomorphic symmetry groups. The classification of finite reflection groups of '''R'''<sup>3</sup> is an instance of the [[ADE classification]].
 
== Kaleidoscopes ==
 
Reflection groups have deep relations with [[kaleidoscope]]s, as discussed in {{Harv|Goodman|2004}}.
 
== Relation with Coxeter groups ==
 
A reflection group ''W'' admits a [[group presentation|presentation]] of a special kind discovered and studied by [[H.S.M. Coxeter]]. The reflections in the faces of a fixed [[fundamental domain|fundamental]] "chamber" are generators ''r''<sub>''i''</sub> of ''W'' of order 2. All relations between them formally follow from the relations
 
: <math>(r_i r_j)^{c_{ij}}=1</math>
 
expressing the fact that the product of the reflections ''r''<sub>''i''</sub> and ''r''<sub>''j''</sub> in two hyperplanes ''H''<sub>''i''</sub> and ''H''<sub>''j''</sub> meeting at an angle <math>\pi/c_{ij}</math> is a [[rotation]] by the angle <math>2\pi/c_{ij}</math> fixing the subspace ''H''<sub>''i''</sub>&nbsp;∩&nbsp;''H''<sub>''j''</sub> of codimension 2. Thus, viewed as an abstract group, every reflection group is a [[Coxeter group]].
 
== Finite fields ==
 
When working over finite fields, one defines a "reflection" as a map that fixes a hyperplane (otherwise for example there would be no reflections in characteristic 2, as <math>-1=1</math> so reflections are the identity).{{Citation needed|date=April 2010}} Geometrically, this amounts to including [[Shear mapping|shear]]s in a hyperplane. Reflection groups over finite fields of characteristic not 2 were classified in {{Harv|Zalesskiĭ|Serežkin|1981}}.
 
== Generalizations ==
Discrete [[isometry group]]s of more general [[Riemannian manifold]]s generated by reflections have also been considered. The most important class arises from [[Riemannian symmetric space]]s of rank 1: the [[n-sphere]] ''S''<sup>''n''</sup>, corresponding to finite reflection groups, the Euclidean space '''R'''<sup>''n''</sup>, corresponding to
affine reflection groups, and the [[hyperbolic space]] ''H''<sup>''n''</sup>, where the corresponding groups are called '''hyperbolic reflection groups'''. In two dimensions, [[triangle group]]s include reflection groups of all three kinds.
 
== See also ==
 
* [[Hyperplane arrangement]]
* [[Chevalley–Shephard–Todd theorem]]
 
== References ==
 
Standard references include {{Harv|Humphreys|1992}} and {{Harv|Grove|Benson|1996}}.
 
{{Reflist}}
{{refbegin}}
* {{Citation
|authorlink=H.S.M. Coxeter
|first=H.S.M.
|last=Coxeter
|title=Discrete groups generated by reflections
|journal=Ann. of Math.
|volume=35
|year=1934
|pages=588–621
}}
* {{Citation
|authorlink=H.S.M. Coxeter
|first=H.S.M.
|last=Coxeter
|title=The complete enumeration of finite groups of the form <math>r_i^2=(r_ir_j)^{k_{ij}}=1</math>
|journal=J. London Math. Soc.
|volume=10
|year=1935
|pages=21–25
}}
*{{Citation
|first=Roe
|last=Goodman
|url=http://www.math.rutgers.edu/~goodman/pub/monthly.pdf
|title=The Mathematics of Mirrors and Kaleidoscopes
|journal=[[American Mathematical Monthly]]
|date=April 2004
}}
* {{Citation
|title=Reflection groups and Coxeter groups
|isbn=978-0-521-43613-7
|first=James E.
|last=Humphreys
|year=1992
|publisher=[[Cambridge University Press]]
}}
* {{Citation
|doi=10.1070/IM1981v017n03ABEH001369
|first1=A E
|last1=Zalesskiĭ
|first2=V N
|last2=Serežkin
|title=Finite Linear Groups Generated by Reflections
|journal=Math. USSR Izv.
|year=1981
|volume=17
|number=3
|pages=477–503
}}
*{{Citation
|url=http://www.cms.math.ca/Publications/Reviews/2003/rev4.pdf
|title=Reflection groups and invariant theory (review)
|first=Richard
|last=Kane
}}
*{{Citation
|arxiv=math/0405135
|title=Jacobians of reflection groups
|first1=Julia
|last1=Hartmann
|first2=Anne V.
|last2=Shepler
}}
*{{Citation
|arxiv=math.AG/0610938
|title=Reflection groups in algebraic geometry
|authorlink=Igor V. Dolgachev
|first=Igor V.
|last=Dolgachev
}}
{{refend}}
 
== External links ==
 
* {{eom|id=Reflection_group|title=Reflection group}}
 
[[Category:Reflection groups| ]]

Latest revision as of 17:51, 12 October 2014

Hello and welcome. My title is Irwin and I completely dig that title. Managing individuals is his profession. The favorite hobby for my children and me is to perform baseball and I'm trying to make it a occupation. North Dakota is exactly where me and my husband live.

Have a look at my homepage - http://www.1a-pornotube.com/