In group theory and geometry, a reflection group is a group which is generated by some set of reflections of a finite-dimensional Euclidean space. Standard references on them include (Humphreys 1992) and (Grove & Benson 1996).
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Examples
Reflection groups include Weyl groups, complex reflection groups, and reflection groups defined over arbitrary fields.
With regard to ordinary reflections in planes in three-dimensional space, a reflection group is an isometry group generated by these reflections.
2 dimensions
In two-dimensions, the finite reflection groups are the dihedral groups, which are generated by reflection in two lines that form an angle of 2π / n and correspond to the Coxeter diagram I2(n). 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.
The infinite reflection groups include the frieze groups 
and 
the wallpaper groups pmm, p3m1, p4m, and p6m, and reflection through two (or more) lines that intersect in an irrational angle.
3 dimensions
The discrete point groups in three dimensions with this property are Cnv, Dnh, and the together three symmetry groups of the 5 Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron); these can be seen as an example of the ADE classification.
Definition
In general, a reflection group is generated by reflections in hyperplanes: that is, by regarding each hyperplane as a mirror for the reflection.
As a definition by Anne V. Shepler states, reflection groups may be defined over arbitrary fields, including finite fields.
If a reflection group consists of finitely many elements, it is called a finite reflection group; if it consists of infinitely many elements, it is called an infinite reflection group.
Applications
Mathematical tools from geometry, topology, algebra, combinatorics, and representation theory are used to study reflection groups. For example, invariant theory (including modular), arrangements of hyperplanes, regular polytopes, Hecke algebras, Coxeter groups, Shephard groups, and braid groups all play a prominent role in investigations on reflection groups. Reflection groups also appear in coding theory, physics, chemistry, and biology. Reflection groups over finite fields are used in Galois geometry, a part of finite geometry.
Kaleidoscopes
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Reflection groups have deep relations with kaleidoscopes, as discussed in (Goodman 2004).
Relation with Coxeter groups
Reflection groups are deeply connected with Coxeter groups. Simply put, Coxeter groups are abstract groups (given via a presentation), while reflection groups are concrete groups (given as subgroups of linear groups 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 ((rirj)k, corresponding to hyperplanes meeting at an angle of π / k, with rirj being of order k abstracting from a rotation by 2π / k).
The abstract group of a reflection group is a Coxeter group, while conversely a reflection groups 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.
Historically, (Coxeter 1934) proved that every reflection group is a Coxeter group (i.e., has a presentation where all relations are of the form 
or (rirj)k), and indeed this paper introduced the notion of a Coxeter group, while (Coxeter 1935) proved that every finite Coxeter group had a representation as a reflection group, and classified finite Coxeter groups.
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 − 1 = 1 so reflections are the identity). Geometrically, this amounts to including shears in a hyperplane. Reflection groups over finite fields of characteristic not 2 were classified in (Zalesskiĭ & Serežkin 1981).
Homology
The first group homology (concretely, the abelianization) of reflection groups is 
for some k, as reflections must map to an order 2 element. For example for the symmetric group this is the sign map.
The second homology, known classically as the Schur multiplier, of finite reflection groups on Euclidean was computed in (Ihara & Yokonuma 1965), while the second homology of infinite reflection groups on Euclidean space was computed in (Yokonuma 1965). A more unified account is given in (Howlett 1988). In all cases, the second homology is (for fundamentally the same reason as for first homology – the group is "at" 2). For each family of group (An, etc.), the rank k stabilizes (is constant after a certain N), while for G2(p) it alternates depending on the parity of p.
See also
References
- Coxeter, H.S.M. (1934), "Discrete groups generated by reflections", Ann. of Math. 35: 588–621
- Coxeter, H.S.M. (1935), "The complete enumeration of finite groups of the form
", J. London Math. Soc. 10: 21–25 - Goodman, Roe (April 2004), "The Mathematics of Mirrors and Kaleidoscopes", American Mathematical Monthly, http://www.math.rutgers.edu/~goodman/pub/monthly.pdf
- Howlett, Robert B. (1988), "On the Schur Multipliers of Coxeter Groups", Journal of the London Mathematical Society, 2 38 (2): 263–276, doi:
- Humphreys, James E. (1992), Reflection groups and Coxeter groups, Cambridge University Press, ISBN 978-0-521-43613-7
- Ihara, S.; Yokonuma, Takeo (1965), "On the second cohomology groups (Schur-multipliers) of finite reflection groups", Jour. Fac. Sci. Univ. Tokyo, Sect. 1 11: 155–171, http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6049/1/jfs110203.pdf
- Yokonuma, Takeo (1965), "On the second cohomology groups (Schur-multipliers) of infinite discrete reflection groups", Jour. Fac. Sci. Univ. Tokyo, Sect. 1 11: 173–186
- Zalesskiĭ, A E; Serežkin, V N (1981), "Finite Linear Groups Generated by Reflections", Math. USSR Izv. 17 (3): 477–503, doi:
- Kane, Richard, Reflection groups and invariant theory (review), http://www.cms.math.ca/Publications/Reviews/2003/rev4.pdf
- Hartmann, Julia; Shepler, Anne V., Jacobians of reflection groups, http://arxiv.org/abs/math/0405135
- Dolgachev, Igor V., Reflection groups in algebraic geometry, http://arxiv.org/abs/math.AG/0610938
External links
- Reflection group at Encyclopaedia of Mathematics, SpringerLink
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