Coxeter group
Encyclopedia
In mathematics
, a Coxeter group, named after H.S.M. Coxeter
, is an abstract group
that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection group
s; the symmetry group
s of regular polyhedra
are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection group
s, and finite Coxeter groups were classified in .
Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry group
s of regular polytope
s, and the Weyl group
s of simple Lie algebras. Examples of infinite Coxeter groups include the triangle group
s corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane
, and the Weyl groups of infinitedimensional Kac–Moody algebra
s.
Standard references include and .
with the presentation
where and for .
The condition means no relation of the form should be imposed.
The pair (W,S) where W is a Coxeter group with generators S={r_{1},...,r_{n}} is called Coxeter system. Note that in general S is not uniquely determined by W. For example, the Coxeter groups of type BC_{3} and A_{1}xA_{3} are isomorphic but the Coxeter systems are not equivalent (see below for an explanation of this notation).
A number of conclusions can be drawn immediately from the above definition.
The Coxeter matrix is the n×n, symmetric matrix with entries m_{i j}. Indeed, every symmetric matrix with positive integer and ∞ entries and with 1's on the diagonal such that all nondiagonal entries are greater than 1 serves to define a Coxeter group.
The Coxeter matrix can be conveniently encoded by a Coxeter graph, as per the following rules.
In particular, two generators commute if and only if they are not connected by an edge.
Furthermore, if a Coxeter graph has two or more connected component
s, the associated group is the direct product
of the groups associated to the individual components.
Thus the disjoint union
of Coxeter graphs yields a direct product
of Coxeter groups.
1 through n are placed in a row with each vertex connected by an unlabelled edge to its immediate neighbors gives rise to the symmetric group
S_{n+1}; the generators
correspond to the transpositions (1 2), (2 3), ... (n n+1). Two nonconsecutive transpositions always commute, while (k k+1) (k+1 k+2) gives the 3cycle (k k+2 k+1). Of course this only shows that S_{n+1} is a quotient group
of the Coxeter group described by the graph, but it is not too difficult to check that equality holds.
s. 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 (, corresponding to hyperplanes meeting at an angle of , with being of order k abstracting from a rotation by ).
The abstract group of a reflection group is a Coxeter group, while conversely a reflection group 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, proved that every reflection group is a Coxeter group (i.e., has a presentation where all relations are of the form or ), and indeed this paper introduced the notion of a Coxeter group, while proved that every finite Coxeter group had a representation as a reflection group, and classified finite Coxeter groups.
s of finitedimensional Euclidean spaces.
The finite Coxeter groups consist of three oneparameter families of increasing rank one oneparameter family of dimension two, and six exceptional
groups: and
can be realized as a Coxeter group. The Weyl groups are the families and and the exceptions and denoted in Weyl group notation as The nonWeyl groups are the exceptions and and the family except where this coincides with one of the Weyl groups (namely and ).
This can be proven by comparing the restrictions on (undirected) Dynkin diagrams with the restrictions on Coxeter diagrams of finite groups: formally, the Coxeter graph can be obtained from the Dynkin diagram by discarding the direction of the edges, and replacing every double edge with an edge labelled 4 and every triple edge by an edge labelled 6. Also note that every finitely generated Coxeter group is an Automatic group
. Dynkin diagrams have the additional restriction that the only permitted edge labels are 2, 3, 4, and 6, which yields the above. Geometrically, this corresponds to the crystallographic restriction theorem
, and the fact that excluded polytopes do not fill space or tile the plane – for the dodecahedron (dually, icosahedron) does not fill space; for the 120cell (dually, 600cell) does not fill space; for a pgon does not tile the plane except for or 6 (the triangular, square, and hexagonal tilings, respectively).
Note further that the (directed) Dynkin diagrams B_{n} and C_{n} give rise to the same Weyl group (hence Coxeter group), because they differ as directed graphs, but agree as undirected graphs – direction matters for root systems but not for the Weyl group; this corresponds to the hypercube and crosspolytope being different regular polytopes but having the same symmetry group.
s of regular polytope
s are finite Coxeter groups. Note that dual polytopes have the same symmetry group.
There are three series of regular polytopes in all dimensions. The symmetry group of a regular nsimplex
is the symmetric group
S_{n+1}, also known as the Coxeter group of type A_{n}. The symmetry group of the ncube
and its dual, the ncrosspolytope
is BC_{n}, and is known as the hyperoctahedral group
.
The exceptional regular polytopes in dimensions two, three, and four, correspond to other Coxeter groups. In two dimensions, the dihedral group
s, which are the symmetry groups of regular polygon
s, form the series I_{2}(p). In three dimensions, the symmetry group of the regular dodecahedron and its dual, the regular icosahedron
, is H_{3}, known as the full icosahedral group. In four dimensions, there are three special regular polytopes, the 24cell, the 120cell, and the 600cell. The first has symmetry group F_{4}, while the other two are dual and have symmetry group H_{4}.
The Coxeter groups of type D_{n}, E_{6}, E_{7}, and E_{8} are the symmetry groups of certain semiregular polytopes
.
The affine Coxeter groups form a second important series of Coxeter groups. These are not finite themselves, but each contains a normal
abelian
subgroup
such that the corresponding quotient group
is finite. In each case, the quotient group is itself a Coxeter group, and the Coxeter graph is obtained from the Coxeter graph of the Coxeter group by adding an additional vertex and one or two additional edges. For example, for n ≥ 2, the graph consisting of n+1 vertices in a circle is obtained from A_{n} in this way, and the corresponding Coxeter group is the affine Weyl group of A_{n}. For n = 2, this can be pictured as the symmetry group of the standard tiling of the plane by equilateral triangles.
A list of the affine Coxeter groups follows:
The subscript is one less than the number of nodes in each case, since each of these groups was obtained by adding a node to a finite group's graph.
, notably including the hyperbolic triangle group
s.
l on a Coxeter group, namely the minimum number of uses of generators required to express a group element; this is precisely the length in the word metric in the Cayley graph
. An expression for v using l(v) generators is a reduced word. For example, the permutation (13) in S_{3} has two reduced words, (12)(23)(12) and (23)(12)(23). The function defines a map generalizing the sign map for the symmetric group.
Using reduced words one may define three partial orders on the Coxeter group, the (left) weak order, the absolute order and the Bruhat order
(named for François Bruhat
). An element v exceeds an element u in the Bruhat order if some (or equivalently, any) reduced word for v contains a reduced word for u as a substring, where some letters (in any position) are dropped. In the weak order, v ≥ u if some reduced word for v contains a reduced word for u as an initial segment. Indeed, the word length makes this into a graded poset
. The Hasse diagram
s corresponding to these orders are objects of study, and are related to the Cayley graph
determined by the generators. The absolute order is defined analogously to the weak order, but with generating set/alphabet consisting of all conjugates of the Coxeter generators.
For example, the permutation (1 2 3) in S_{3} has only one reduced word, (12)(23), so covers (12) and (23) in the Bruhat order but only covers (12) in the weak order.
, i.e. it is isomorphic to the direct sum of several copies of the cyclic group
Z_{2}. This may be restated in terms of the first homology group of W.
The Schur multiplier
M(W) (related to the second homology) was computed in for finite reflection groups and in for affine reflection groups, with a more unified account given in . In all cases, the Schur multiplier is also an elementary abelian 2group. For each infinite family {W_{n}} of finite or affine Weyl groups, the rank of M(W) stabilizes as n goes to infinity.
Further reading1
journal=J. London Math. Soc.
volume=10
year=1935
pages=21–25
}}
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Coxeter group, named after H.S.M. Coxeter
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, was a Britishborn Canadian geometer. Coxeter is regarded as one of the great geometers of the 20th century. He was born in London but spent most of his life in Canada....
, is an abstract group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection group
Reflection group
In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finitedimensional Euclidean space. 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...
s; the symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
s of regular polyhedra
Regular polyhedron
A regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edgetransitive, vertextransitive and facetransitive  i.e. it is transitive on its flags...
are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection group
Reflection group
In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finitedimensional Euclidean space. 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...
s, and finite Coxeter groups were classified in .
Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
s of regular polytope
Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or jfaces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
s, and the Weyl group
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
s of simple Lie algebras. Examples of infinite Coxeter groups include the triangle group
Triangle group
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle...
s corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane
Hyperbolic space
In mathematics, hyperbolic space is a type of nonEuclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
, and the Weyl groups of infinitedimensional Kac–Moody algebra
Kac–Moody algebra
In mathematics, a Kac–Moody algebra is a Lie algebra, usually infinitedimensional, that can be defined by generators and relations through a generalized Cartan matrix...
s.
Standard references include and .
Definition
Formally, a Coxeter group can be defined as a groupGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
with the presentation
Presentation of a group
In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...
where and for .
The condition means no relation of the form should be imposed.
The pair (W,S) where W is a Coxeter group with generators S={r_{1},...,r_{n}} is called Coxeter system. Note that in general S is not uniquely determined by W. For example, the Coxeter groups of type BC_{3} and A_{1}xA_{3} are isomorphic but the Coxeter systems are not equivalent (see below for an explanation of this notation).
A number of conclusions can be drawn immediately from the above definition.
 The relation m_{i i} = 1 means that (r_{i}r_{i} )^{1} = (r_{i} )^{2} = 1 for all i ; the generators are involutions.
 If m_{i j} = 2, then the generators r_{i} and r_{j} commute. This follows by observing that

 xx = yy = 1,
 together with
 xyxy = 1
 implies that
 xy = x(xyxy)y = (xx)yx(yy) = yx.
 Alternatively, since the generators are involutions, , so , and thus is equal to the commutatorCommutatorIn mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.Group theory:...
. In order to avoid redundancy among the relations, it is necessary to assume that m_{i j} = m_{j i}. This follows by observing that
 yy = 1,
 together with
 (xy)^{m} = 1
 implies that
 (yx)^{m} = (yx)^{m}yy = y(xy)^{m}y = yy = 1.
 Alternatively, and are conjugate elements, as .
The Coxeter matrix is the n×n, symmetric matrix with entries m_{i j}. Indeed, every symmetric matrix with positive integer and ∞ entries and with 1's on the diagonal such that all nondiagonal entries are greater than 1 serves to define a Coxeter group.
The Coxeter matrix can be conveniently encoded by a Coxeter graph, as per the following rules.
 The vertices of the graph are labelled by generator subscripts.
 Vertices i and j are connected if and only if m_{i j} ≥ 3.
 An edge is labelled with the value of m_{i j} whenever it is 4 or greater.
In particular, two generators commute if and only if they are not connected by an edge.
Furthermore, if a Coxeter graph has two or more connected component
Connected component (graph theory)
In graph theory, a connected component of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices. For example, the graph shown in the illustration on the right has three connected components...
s, the associated group is the direct product
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
of the groups associated to the individual components.
Thus the disjoint union
Disjoint union
In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
of Coxeter graphs yields a direct product
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
of Coxeter groups.
An example
The graph in which verticesVertex (graph theory)
In graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs...
1 through n are placed in a row with each vertex connected by an unlabelled edge to its immediate neighbors gives rise to the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
S_{n+1}; the generators
Generating set of a group
In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
correspond to the transpositions (1 2), (2 3), ... (n n+1). Two nonconsecutive transpositions always commute, while (k k+1) (k+1 k+2) gives the 3cycle (k k+2 k+1). Of course this only shows that S_{n+1} is a quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
of the Coxeter group described by the graph, but it is not too difficult to check that equality holds.
Connection with reflection groups
Coxeter groups are deeply connected with reflection groupReflection group
In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finitedimensional Euclidean space. 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...
s. 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 (, corresponding to hyperplanes meeting at an angle of , with being of order k abstracting from a rotation by ).
The abstract group of a reflection group is a Coxeter group, while conversely a reflection group 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, proved that every reflection group is a Coxeter group (i.e., has a presentation where all relations are of the form or ), and indeed this paper introduced the notion of a Coxeter group, while proved that every finite Coxeter group had a representation as a reflection group, and classified finite Coxeter groups.
Finite Coxeter groups
Classification
The finite Coxeter groups were classified in , in terms of Coxeter–Dynkin diagrams; they are all represented by reflection groupReflection group
In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finitedimensional Euclidean space. 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...
s of finitedimensional Euclidean spaces.
The finite Coxeter groups consist of three oneparameter families of increasing rank one oneparameter family of dimension two, and six exceptional
Exceptional object
Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects as well as a finite number of exceptions that don't fit into any series. These are known as exceptional...
groups: and
Weyl groups
Many, but not all of these, are Weyl groups, and every Weyl groupWeyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
can be realized as a Coxeter group. The Weyl groups are the families and and the exceptions and denoted in Weyl group notation as The nonWeyl groups are the exceptions and and the family except where this coincides with one of the Weyl groups (namely and ).
This can be proven by comparing the restrictions on (undirected) Dynkin diagrams with the restrictions on Coxeter diagrams of finite groups: formally, the Coxeter graph can be obtained from the Dynkin diagram by discarding the direction of the edges, and replacing every double edge with an edge labelled 4 and every triple edge by an edge labelled 6. Also note that every finitely generated Coxeter group is an Automatic group
Automatic group
In mathematics, an automatic group is a finitely generated group equipped with several finitestate automata. These automata can tell if a given word representation of a group element is in a "canonical form" and can tell if two elements given in canonical words differ by a generator.More...
. Dynkin diagrams have the additional restriction that the only permitted edge labels are 2, 3, 4, and 6, which yields the above. Geometrically, this corresponds to the crystallographic restriction theorem
Crystallographic restriction theorem
The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2fold, 3fold, 4fold, and 6fold...
, and the fact that excluded polytopes do not fill space or tile the plane – for the dodecahedron (dually, icosahedron) does not fill space; for the 120cell (dually, 600cell) does not fill space; for a pgon does not tile the plane except for or 6 (the triangular, square, and hexagonal tilings, respectively).
Note further that the (directed) Dynkin diagrams B_{n} and C_{n} give rise to the same Weyl group (hence Coxeter group), because they differ as directed graphs, but agree as undirected graphs – direction matters for root systems but not for the Weyl group; this corresponds to the hypercube and crosspolytope being different regular polytopes but having the same symmetry group.
Properties
Some properties of the finite Coxeter groups are given in the following table:Group symbol 
Alternate symbol 
Bracket notation Coxeter notation In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M... 
Rank  Order Order (group theory) In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements.... 
Related polytopes Uniform polytope A uniform polytope is a vertextransitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons.... 
CoxeterDynkin diagram CoxeterDynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... 

A_{n}  A_{n}  [3^{n}]  n  (n + 1)!  nsimplex Simplex In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an nsimplex is an ndimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2simplex is a triangle, a 3simplex is a tetrahedron,... 
.. 
BC_{n}  C_{n}  [4,3^{n1}]  n  2^{n} n!  nhypercube Hypercube In geometry, a hypercube is an ndimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An... / ncrosspolytope Crosspolytope In geometry, a crosspolytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a crosspolytope are all the permutations of . The crosspolytope is the convex hull of its vertices... 
... 
D_{n}  B_{n}  [3^{n3,1,1}]  n  2^{n−1} n!  ndemihypercube  ... 
E_{6}  E_{6}  [3^{2,2,1}]  6  72x6! = 51840  2_{21}, 1_{22}  
E_{7}  E_{7}  [3^{3,2,1}]  7  72x8! = 2903040  3_{21}, 2_{31}, 1_{32}  
E_{8} E8 (mathematics) In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8... 
E_{8}  [3^{4,2,1}]  8  192x10! = 696729600  4_{21}, 2_{41}, 1_{42}  
F_{4}  F_{4}  [3,4,3]  4  1152  24cell  
G_{2}    [6]  2  12  hexagon  
H_{2}  G_{2}  [5]  2  10  pentagon Pentagon In geometry, a pentagon is any fivesided polygon. A pentagon may be simple or selfintersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a selfintersecting pentagon. Regular pentagons :In a regular pentagon, all sides are equal in length and... 

H_{3}  G_{3}  [3,5]  3  120  icosahedron Icosahedron In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids.... / dodecahedron 

H_{4}  G_{4}  [3,3,5]  4  14400  120cell / 600cell  
I_{2}(p)  D_{2}^{p}  [p]  2  2p  pgon Regular polygon A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:... 
Symmetry groups of regular polytopes
All symmetry groupSymmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
s of regular polytope
Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or jfaces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
s are finite Coxeter groups. Note that dual polytopes have the same symmetry group.
There are three series of regular polytopes in all dimensions. The symmetry group of a regular nsimplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an nsimplex is an ndimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2simplex is a triangle, a 3simplex is a tetrahedron,...
is the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
S_{n+1}, also known as the Coxeter group of type A_{n}. The symmetry group of the ncube
Cube
In geometry, a cube is a threedimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
and its dual, the ncrosspolytope
Crosspolytope
In geometry, a crosspolytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a crosspolytope are all the permutations of . The crosspolytope is the convex hull of its vertices...
is BC_{n}, and is known as the hyperoctahedral group
Hyperoctahedral group
In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a crosspolytope. Groups of this type are identified by a parameter n, the dimension of the hypercube....
.
The exceptional regular polytopes in dimensions two, three, and four, correspond to other Coxeter groups. In two dimensions, the dihedral group
Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
s, which are the symmetry groups of regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:...
s, form the series I_{2}(p). In three dimensions, the symmetry group of the regular dodecahedron and its dual, the regular icosahedron
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
, is H_{3}, known as the full icosahedral group. In four dimensions, there are three special regular polytopes, the 24cell, the 120cell, and the 600cell. The first has symmetry group F_{4}, while the other two are dual and have symmetry group H_{4}.
The Coxeter groups of type D_{n}, E_{6}, E_{7}, and E_{8} are the symmetry groups of certain semiregular polytopes
Thorold Gosset
Thorold Gosset was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher.According to H. S. M...
.
Affine Coxeter groups
The affine Coxeter groups form a second important series of Coxeter groups. These are not finite themselves, but each contains a normal
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
such that the corresponding quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
is finite. In each case, the quotient group is itself a Coxeter group, and the Coxeter graph is obtained from the Coxeter graph of the Coxeter group by adding an additional vertex and one or two additional edges. For example, for n ≥ 2, the graph consisting of n+1 vertices in a circle is obtained from A_{n} in this way, and the corresponding Coxeter group is the affine Weyl group of A_{n}. For n = 2, this can be pictured as the symmetry group of the standard tiling of the plane by equilateral triangles.
A list of the affine Coxeter groups follows:
Group symbol 
Witt Ernst Witt Ernst Witt was a German mathematician born on the island of Als . Shortly after his birth, he and his parents moved to China, and he did not return to Europe until he was nine.... symbol 
Bracket notation Coxeter notation In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M... 
Related uniform tessellation(s)  CoxeterDynkin diagram CoxeterDynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... 

P_{n+1}  [3^{[n+1]}]  Simplectic honeycomb :Triangular tiling :Tetrahedraloctahedral honeycomb Tetrahedraloctahedral honeycomb The tetrahedraloctahedral honeycomb or alternated cubic honeycomb is a spacefilling tessellation in Euclidean 3space. It is composed of alternating octahedra and tetrahedra in a ratio of 1:2.... 
...  
S_{n+1}  [4,3^{n2},3^{1,1}]  Demihypercubic honeycomb  ...  
R_{n+1}  [4,3^{n1},4]  Hypercubic honeycomb  ...  
Q_{n+1}  [ 3^{1,1},3^{n3},3^{1,1}]  Demihypercubic honeycomb  ...  
T_{7}  [3^{2,2,2}]  2_{22}  
T_{8}  [3^{3,3,1}]  3_{31}, 1_{33}  
T_{9}  [3^{5,2,1}]  5_{21} 5 21 honeycomb In geometry, the 521 honeycomb is a uniform tessellation of 8dimensional Euclidean space.This honeycomb was first studied by Gosset who called it a 9ic semiregular figure... , 2_{51}, 1_{52} 

U_{5}  [3,4,3,3]  16cell honeycomb 24cell honeycomb 

V_{3}  [6,3]  Hexagonal tiling and Triangular tiling 

W_{2}  [∞]  apeirogon Apeirogon An apeirogon is a degenerate polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides.Like any polygon, it is a sequence of line segments and angles... 
The subscript is one less than the number of nodes in each case, since each of these groups was obtained by adding a node to a finite group's graph.
Hyperbolic Coxeter groups
There are infinitely many hyperbolic Coxeter groups describing reflection groups in hyperbolic spaceHyperbolic geometry
In mathematics, hyperbolic geometry is a nonEuclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
, notably including the hyperbolic triangle group
Triangle group
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle...
s.
Partial orders
A choice of reflection generators gives rise to a length functionLength function
In mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group.Definition:...
l on a Coxeter group, namely the minimum number of uses of generators required to express a group element; this is precisely the length in the word metric in the Cayley graph
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem and uses a specified, usually finite, set of generators for the group...
. An expression for v using l(v) generators is a reduced word. For example, the permutation (13) in S_{3} has two reduced words, (12)(23)(12) and (23)(12)(23). The function defines a map generalizing the sign map for the symmetric group.
Using reduced words one may define three partial orders on the Coxeter group, the (left) weak order, the absolute order and the Bruhat order
Bruhat order
In mathematics, the Bruhat order is a partial order on the elements of a Coxeter group, that corresponds to the inclusion order on Schubert varieties.History:The Bruhat order on the Schubert varieties of a flag manifold or Grassmannian...
(named for François Bruhat
François Bruhat
François Georges René Bruhat was a French mathematician who worked on algebraic groups. The Bruhat order of a Weyl group, the Bruhat decomposition, and the Schwartz–Bruhat functions are named after him....
). An element v exceeds an element u in the Bruhat order if some (or equivalently, any) reduced word for v contains a reduced word for u as a substring, where some letters (in any position) are dropped. In the weak order, v ≥ u if some reduced word for v contains a reduced word for u as an initial segment. Indeed, the word length makes this into a graded poset
Graded poset
In mathematics, in the branch of combinatorics, a graded poset, sometimes called a ranked poset , is a partially ordered set P equipped with a rank function ρ from P to N compatible with the ordering such that whenever y covers x, then...
. The Hasse diagram
Hasse diagram
In order theory, a branch of mathematics, a Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction...
s corresponding to these orders are objects of study, and are related to the Cayley graph
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem and uses a specified, usually finite, set of generators for the group...
determined by the generators. The absolute order is defined analogously to the weak order, but with generating set/alphabet consisting of all conjugates of the Coxeter generators.
For example, the permutation (1 2 3) in S_{3} has only one reduced word, (12)(23), so covers (12) and (23) in the Bruhat order but only covers (12) in the weak order.
Homology
Since a Coxeter group W is generated by finitely many elements of order 2, its abelianization is an elementary abelian 2groupElementary Abelian group
In group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order p, where p is a prime; in particular it is a pgroup....
, i.e. it is isomorphic to the direct sum of several copies of the cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .Definition:A group G is called cyclic if there exists an element g...
Z_{2}. This may be restated in terms of the first homology group of W.
The Schur multiplier
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.Examples and properties:...
M(W) (related to the second homology) was computed in for finite reflection groups and in for affine reflection groups, with a more unified account given in . In all cases, the Schur multiplier is also an elementary abelian 2group. For each infinite family {W_{n}} of finite or affine Weyl groups, the rank of M(W) stabilizes as n goes to infinity.
See also
 Artin group
 Triangle groupTriangle groupIn mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle...
 Coxeter element
 Coxeter numberCoxeter numberIn mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group, hence also of a root system or its Weyl group. It is named after H.S.M. Coxeter.Definitions:...
 Complex reflection groupComplex reflection groupIn mathematics, a complex reflection group is a group acting on a finitedimensional complex vector space, that is generated by complex reflections: nontrivial elements that fix a complex hyperplane in space pointwise...
 Chevalley–Shephard–Todd theoremChevalley–Shephard–Todd theoremIn mathematics, the Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring of invariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group is generated by pseudoreflections. In the case of subgroups of the complex...
 Hecke algebraHecke algebraIn mathematics, the Iwahori–Hecke algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a oneparameter deformation of the group algebra of a Coxeter group....
, a quantum deformation of the group algebraGroup algebraIn mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...  Kazhdan–Lusztig polynomialKazhdan–Lusztig polynomialIn representation theory, a Kazhdan–Lusztig polynomial Py,w is a member of a family of integral polynomials introduced by . They are indexed by pairs of elements y, w of a Coxeter group W, which can in particular be the Weyl group of a Lie group. Motivation and history:In the spring of 1978...
 Longest element of a Coxeter groupLongest element of a Coxeter groupIn mathematics, the longest element of a Coxeter group is the unique element of maximal length in a finite Coxeter group with respect to the chosen generating set consisting of simple reflections. It is often denoted by w0...
Further reading1
journal=J. London Math. Soc.
volume=10
year=1935
pages=21–25
}}
 Larry C Grove and Clark T. Benson, Finite Reflection Groups, Graduate texts in mathematics, vol. 99, Springer, (1985)
 James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29 (1990)
 Richard Kane, Reflection Groups and Invariant Theory, CMS Books in Mathematics, Springer (2001)
 Anders BjörnerAnders BjörnerAnders Björner is a Swedish professor of mathematics, Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden...
and Francesco Brenti, Combinatorics of Coxeter Groups, Graduate Texts in MathematicsGraduate Texts in MathematicsGraduate Texts in Mathematics is a series of graduatelevel textbooks in mathematics published by SpringerVerlag. The books in this series, like the other SpringerVerlag mathematics series, are yellow books of a standard size...
, vol. 231, Springer, (2005)  Howard Hiller, Geometry of Coxeter groups. Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.London, 1982. iv+213 pp. ISBN 0273085174
 Nicolas Bourbaki, Lie Groups and Lie Algebras: Chapter 46, Elements of Mathematics, Springer (2002). ISBN 9783540426509