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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 ==
=== Two dimensions ===

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, nor contain any – they are 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.

=== Three dimensions ===

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]].

== 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]].<ref>{{harvs|last=Coxeter|year1=1934|year2=1935|txt}}</ref> 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. Geometrically, this amounts to including [[Shear mapping|shear]]s in a hyperplane. Reflection groups over finite fields of characteristic not 2 were classified by {{Harvtxt|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]]
* Reflection groups are related to [[kaleidoscope]]s.{{sfnp|Goodman|2004}}
* [[Parabolic subgroup of a reflection group]]

== References ==
===Notes===
{{Reflist}}

===Bibliography===
{{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
|issue=3
|year=1934
|pages=588–621
|doi=10.2307/1968753
|jstor=1968753
|citeseerx=10.1.1.128.471
}}
* {{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
|doi=10.1112/jlms/s1-10.37.21
}}
*{{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]]
|volume=111
|issue=4
|pages=281–298
|date=April 2004
|doi=10.2307/4145238
|jstor=4145238
|citeseerx=10.1.1.127.6227
}}
* {{Citation
|doi=10.1070/IM1981v017n03ABEH001369
|first1=Aleksandr 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
|bibcode=1981IzMat..17..477Z
}}
{{refend}}

===Textbooks===
{{refbegin}}
*{{citation
  | last1 = Borovik  | first1 = Alexandre | author1-link = Alexandre Borovik
  | last2 = Borovik  | first2 = Anna
  | title = Mirrors and reflections : the geometry of finite reflection groups
  | title-link = Mirrors and Reflections
  | publisher = [[Springer Publishing|Springer]]
  | location = New York
  | year = 2010
  | isbn = 9780387790664}}
*{{citation
 | last1 = Grove | first1 = L. C.
 | last2 = Benson | first2 = C. T.
 | doi = 10.1007/978-1-4757-1869-0
 | edition = 2nd
 | isbn = 0-387-96082-1
 | mr = 777684
 | publisher = Springer-Verlag, New York
 | series = Graduate Texts in Mathematics
 | title = Finite reflection groups
 | volume = 99
 | year = 1985}}
* {{Citation
|title=Reflection groups and Coxeter groups
|isbn=978-0-521-43613-7
|first=James E.
|last=Humphreys
|authorlink=James E. Humphreys
|year=1992
|publisher=[[Cambridge University Press]]
|url=https://books.google.com/books?id=ODfjmOeNLMUC
}}
{{refend}}


==External links==
*{{Commonscatinline|Reflection groups}}
*{{eom|id=Reflection_group|title=Reflection group}}

[[Category:Coxeter groups|*]]
[[Category:Reflection groups| ]]