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| {{For|the Reflection Group On the Future of Europe|European Council}}
| | 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.<br><br>Have a look at my homepage - [http://www.1a-pornotube.com/blog/84958 http://www.1a-pornotube.com/] |
| 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.
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| == Definition ==
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| 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).
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| 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]].
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| == Examples ==
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| === Plane ===
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| 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.
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| 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.
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| === Space ===
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| 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]].
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| == Kaleidoscopes ==
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| Reflection groups have deep relations with [[kaleidoscope]]s, as discussed in {{Harv|Goodman|2004}}.
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| == Relation with Coxeter groups ==
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| 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
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| : <math>(r_i r_j)^{c_{ij}}=1</math>
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| 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> ∩ ''H''<sub>''j''</sub> of codimension 2. Thus, viewed as an abstract group, every reflection group is a [[Coxeter group]].
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| == Finite fields ==
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| 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}}.
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| == Generalizations ==
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| 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
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| 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.
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| == See also ==
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| * [[Hyperplane arrangement]]
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| * [[Chevalley–Shephard–Todd theorem]]
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| == References ==
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| Standard references include {{Harv|Humphreys|1992}} and {{Harv|Grove|Benson|1996}}.
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| {{Reflist}}
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| {{refbegin}}
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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=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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| |pages=588–621
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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
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| |first=Roe
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| |last=Goodman
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| |url=http://www.math.rutgers.edu/~goodman/pub/monthly.pdf
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| |title=The Mathematics of Mirrors and Kaleidoscopes
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| |journal=[[American Mathematical Monthly]]
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| |date=April 2004
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| }}
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| * {{Citation
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| |title=Reflection groups and Coxeter groups
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| |isbn=978-0-521-43613-7
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| |first=James E.
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| |last=Humphreys
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| |year=1992
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| |publisher=[[Cambridge University Press]]
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| }}
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| * {{Citation
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| |doi=10.1070/IM1981v017n03ABEH001369
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| |first1=A E
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| |last1=Zalesskiĭ
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| |first2=V N
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| |last2=Serežkin
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| |title=Finite Linear Groups Generated by Reflections
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| |journal=Math. USSR Izv.
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| |year=1981
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| |volume=17
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| |number=3
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| |pages=477–503
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| }}
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| *{{Citation
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| |url=http://www.cms.math.ca/Publications/Reviews/2003/rev4.pdf
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| |title=Reflection groups and invariant theory (review)
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| |first=Richard
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| |last=Kane
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| }}
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| *{{Citation
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| |arxiv=math/0405135
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| |title=Jacobians of reflection groups
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| |first1=Julia
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| |last1=Hartmann
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| |first2=Anne V.
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| |last2=Shepler
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| }}
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| *{{Citation
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| |arxiv=math.AG/0610938
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| |title=Reflection groups in algebraic geometry
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| |authorlink=Igor V. Dolgachev
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| |first=Igor V.
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| |last=Dolgachev
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| }}
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| {{refend}}
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| == External links ==
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| * {{eom|id=Reflection_group|title=Reflection group}}
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| [[Category:Reflection groups| ]]
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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/