In
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
and
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a reflection group is a
discrete group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...
which is generated by a set of
reflections of a finite-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
. The symmetry group of a
regular polytope or of a
tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...
s and crystallographic
Coxeter groups. While the
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
is generated by reflections (by the
Cartan–Dieudonné theorem
In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an ''n''-dimensional symmetric bilinear space can be described as the composition of at most ''n'' ...
), it is a continuous group (indeed,
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
), not a discrete group, and is generally considered separately.
Definition
Let ''E'' be a finite-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
. A finite reflection group is a subgroup of the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
of ''E'' which is generated by a set of orthogonal
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
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
fields, leading to
complex reflection groups and analogues of reflection groups over a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
.
Examples
Plane
In two dimensions, the finite reflection groups are the
dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
s, which are generated by reflection in two lines that form an angle of
and correspond to the
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
Conversely, the cyclic
point groups in two dimensions
In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O(2), including O(2) itself. Its ele ...
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 groups
and
and the
wallpaper group
A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformati ...
s
,
,
,
and
. 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 geometry, a point group is a mathematical group of symmetry operations ( isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every ...
''C
nv'', ''D
nh'', and the
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
s of the five
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
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
3 is an instance of the
ADE classification
In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, r ...
.
Relation with Coxeter groups
A reflection group ''W'' admits a
presentation
A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
of a special kind discovered and studied by
H. S. M. Coxeter.
The reflections in the faces of a fixed
fundamental "chamber" are generators ''r''
''i'' of ''W'' of order 2. All relations between them formally follow from the relations
:
expressing the fact that the product of the reflections ''r''
''i'' and ''r''
''j'' in two hyperplanes ''H''
''i'' and ''H''
''j'' meeting at an angle
is a
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
by the angle
fixing the subspace ''H''
''i'' ∩ ''H''
''j'' 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
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 by .
Generalizations
Discrete
isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...
s of more general
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
s generated by reflections have also been considered. The most important class arises from
Riemannian symmetric spaces of rank 1: the
n-sphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, call ...
''S''
''n'', corresponding to finite reflection groups, the Euclidean space R
''n'', corresponding to
affine reflection groups, and the
hyperbolic space ''H''
''n'', where the corresponding groups are called hyperbolic reflection groups. In two dimensions,
triangle groups include reflection groups of all three kinds.
See also
*
Hyperplane arrangement
*
Chevalley–Shephard–Todd theorem
* Reflection groups are related to
kaleidoscopes.
References
Notes
Bibliography
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Textbooks
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External links
*
*{{eom, id=Reflection_group, title=Reflection group
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