In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups,^{[1]} and finite Coxeter groups were classified in 1935.^{[2]}
Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinitedimensional Kac–Moody algebras.^{[3]}^{[4]}^{[5]}
Formally, a Coxeter group can be defined as a group with the presentation
where and is either an integer or for . Here, the condition means that no relation of the form for any integer should be imposed.
The pair where is a Coxeter group with generators is called a Coxeter system. Note that in general is not uniquely determined by . For example, the Coxeter groups of type and are isomorphic but the Coxeter systems are not equivalent, since the former has 3 generators and the latter has 1 + 3 = 4 generators (see below for an explanation of this notation).
A number of conclusions can be drawn immediately from the above definition.
The reason that for is stipulated in the definition is that
together with
already implies that
An alternative proof of this implication is the observation that and are conjugates: indeed .
The Coxeter matrix is the symmetric matrix with entries . Indeed, every symmetric matrix with diagonal entries exclusively 1 and nondiagonal entries in the set is a Coxeter matrix.
The Coxeter matrix can be conveniently encoded by a Coxeter diagram, as per the following rules.
In particular, two generators commute if and only if they are not joined by an edge. Furthermore, if a Coxeter graph has two or more connected components, 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.
The Coxeter matrix, , is related to the Schläfli matrix with entries , but the elements are modified, being proportional to the dot product of the pairwise generators. The Schläfli matrix is useful because its eigenvalues determine whether the Coxeter group is of finite type (all positive), affine type (all nonnegative, at least one zero), or indefinite type (otherwise). The indefinite type is sometimes further subdivided, e.g. into hyperbolic and other Coxeter groups. However, there are multiple nonequivalent definitions for hyperbolic Coxeter groups.
Coxeter group  A_{1}×A_{1}  A_{2}  B_{2}  H_{2}  G_{2}  A_{3}  B_{3}  D_{4}  

Coxeter diagram  
Coxeter matrix  
Schläfli matrix 
The graph A_{n} in which vertices 1 through n are placed in a row with each vertex joined by an unlabelled edge to its immediate neighbors is the Coxeter diagram of the symmetric group S_{n+1}; the generators correspond to the transpositions (1 2), (2 3), ... , (n n+1). Any two nonconsecutive transpositions commute, while multiplying two consecutive transpositions gives a 3cycle : (k k+1) (k+1 k+2) = (k k+2 k+1). Therefore S_{n+1} is a quotient of the Coxeter group having Coxeter diagram A_{n}. Further arguments show that this quotient map is an isomorphism.
Coxeter groups are an abstraction of reflection groups. Coxeter groups are abstract groups, in the sense of being given via a presentation. On the other hand, reflection groups are concrete, in the sense that each of its elements is the composite of finitely many geometric reflections about linear hyperplanes in some euclidean space. Technically, a reflection group is a subgroup of a linear group (or various generalizations) generated by orthogonal matrices of determinant 1. Each generator of a Coxeter group has order 2, which abstracts the geometric fact that performing a reflection twice is the identity. Each relation of the form , corresponding to the geometric fact that, given two hyperplanes meeting at an angle of , the composite of the two reflections about these hyperplanes is a rotation by , which has order k.
In this way, every reflection group may be presented as a Coxeter group.^{[1]} The converse is partially true: every finite Coxeter group admits a faithful representation as a finite reflection group of some Euclidean space.^{[2]} However, not every infinite Coxeter group admits a representation as a reflection group.
Finite Coxeter groups have been classified.^{[2]}
Finite Coxeter groups are classified in terms of their Coxeter diagrams.^{[2]}
The finite Coxeter groups with connected Coxeter diagrams consist of 3 oneparameter families of increasing rank a oneparameter family of dimension two, and 6 exceptional groups: and . Every finite Coxeter group is the direct product of finitely many Coxeter groups in the above list.
Many, but not all of these, are Weyl groups, and every Weyl group 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 ones are the exceptions and and those members of the family that are not exceptionally isomorphic to a Weyl group (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.^{[6]} 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 (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.
Some properties of the finite irreducible Coxeter groups are given in the following table. The order of a reducible group can be computed by the product of its irreducible subgroup orders.
Rank n 
Group symbol 
Alternate symbol 
Bracket notation 
Coxeter graph 
Reflections m = 1⁄2nh^{[7]} 
Coxeter number h 
Order  Group structure^{[8]}  Related polytopes 

1  A_{1}  A_{1}  [ ]  1  2  2  { }  
2  A_{2}  A_{2}  [3]  3  3  6  {3}  
3  A_{3}  A_{3}  [3,3]  6  4  24  {3,3}  
4  A_{4}  A_{4}  [3,3,3]  10  5  120  {3,3,3}  
5  A_{5}  A_{5}  [3,3,3,3]  15  6  720  {3,3,3,3}  
n  A_{n}  A_{n}  [3^{n−1}]  ...  n(n + 1)/2  n + 1  (n + 1)!  nsimplex  
2  B_{2}  C_{2}  [4]  4  4  8  {4}  
3  B_{3}  C_{3}  [4,3]  9  6  48  {4,3} / {3,4}  
4  B_{4}  C_{4}  [4,3,3]  16  8  384  {4,3,3} / {3,3,4}  
5  B_{5}  C_{5}  [4,3,3,3]  25  10  3840  {4,3,3,3} / {3,3,3,4}  
n  B_{n}  C_{n}  [4,3^{n−2}]  ...  n^{2}  2n  2^{n} n!  ncube / northoplex  
4  D_{4}  B_{4}  [3^{1,1,1}]  12  6  192  h{4,3,3} / {3,3^{1,1}}  
5  D_{5}  B_{5}  [3^{2,1,1}]  20  8  1920  h{4,3,3,3} / {3,3,3^{1,1}}  
n  D_{n}  B_{n}  [3^{n−3,1,1}]  ...  n(n − 1)  2(n − 1)  2^{n−1} n!  ndemicube / northoplex  
6  E_{6}  E_{6}  [3^{2,2,1}]  36  12  51840 (72x6!) 


7  E_{7}  E_{7}  [3^{3,2,1}]  63  18  2903040 (72x8!)  3_{21}, 2_{31}, 1_{32}  
8  E_{8}  E_{8}  [3^{4,2,1}]  120  30  696729600 (192x10!)  4_{21}, 2_{41}, 1_{42}  
4  F_{4}  F_{4}  [3,4,3]  24  12  1152  {3,4,3}  
2  G_{2}  – (D^{6} _{2}) 
[6]  6  6  12  {6}  
2  H_{2}  G_{2}  [5]  5  5  10  {5}  
3  H_{3}  G_{3}  [3,5]  15  10  120  {3,5} / {5,3}  
4  H_{4}  G_{4}  [3,3,5]  60  30  14400  ^{[a]}  {5,3,3} / {3,3,5}  
2  I_{2}(n)  D^{n} _{2} 
[n]  n  n  2n 
when n = p^{k} + 1, p prime when n = p^{k} − 1, p prime 
{p} 
The symmetry group of every regular polytope is a finite Coxeter group. 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 B_{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 groups, which are the symmetry groups of regular polygons, form the series I_{2}(p), for p ≥ 3. 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 exceptional 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.
Table of irreducible polytope families  

Family n 
nsimplex  nhypercube  northoplex  ndemicube  1_{k2}  2_{k1}  k_{21}  pentagonal polytope  
Group  A_{n}  B_{n} 


H_{n}  
2  pgon (example: p=7) 
Hexagon 
Pentagon  
3  Tetrahedron 
Cube 
Octahedron 
Tetrahedron 
Dodecahedron 
Icosahedron  
4  5cell 
16cell 
24cell 
120cell 
600cell  
5  5simplex 
5cube 
5orthoplex 
5demicube 

6  6simplex 
6cube 
6orthoplex 
6demicube 
1_{22} 
2_{21} 

7  7simplex 
7cube 
7orthoplex 
7demicube 
1_{32} 
2_{31} 
3_{21} 

8  8simplex 
8cube 
8orthoplex 
8demicube 
1_{42} 
2_{41} 
4_{21} 

9  9simplex 
9cube 
9orthoplex 
9demicube 

10  10simplex 
10cube 
10orthoplex 
10demicube 
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 of the affine Coxeter group is obtained from the Coxeter graph of the quotient group by adding another 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} (the affine symmetric group). For n = 2, this can be pictured as a subgroup of the symmetry group of the standard tiling of the plane by equilateral triangles.
In general, given a root system, one can construct the associated Stiefel diagram, consisting of the hyperplanes orthogonal to the roots along with certain translates of these hyperplanes. The affine Coxeter group (or affine Weyl group) is then the group generated by the (affine) reflections about all the hyperplanes in the diagram.^{[9]} The Stiefel diagram divides the plane into infinitely many connected components called alcoves, and the affine Coxeter group acts freely and transitively on the alcoves, just as the ordinary Weyl group acts freely and transitively on the Weyl chambers. The figure at right illustrates the Stiefel diagram for the root system.
Suppose is an irreducible root system of rank and let be a collection of simple roots. Let, also, denote the highest root. Then the affine Coxeter group is generated by the ordinary (linear) reflections about the hyperplanes perpendicular to , together with an affine reflection about a translate of the hyperplane perpendicular to . The Coxeter graph for the affine Weyl group is the Coxeter–Dynkin diagram for , together with one additional node associated to . In this case, one alcove of the Stiefel diagram may be obtained by taking the fundamental Weyl chamber and cutting it by a translate of the hyperplane perpendicular to .^{[10]}
A list of the affine Coxeter groups follows:
Group symbol 
Witt symbol 
Bracket notation  Coxeter graph 
Related uniform tessellation(s) 

[3^{[n]}]  ... or ... 
Simplectic honeycomb  
[4,3^{n − 3},3^{1,1}]  ...  Demihypercubic honeycomb  
[4,3^{n−2},4]  ...  Hypercubic honeycomb  
[ 3^{1,1},3^{n−4},3^{1,1}]  ...  Demihypercubic honeycomb  
[3^{2,2,2}]  or  2_{22}  
[3^{3,3,1}]  or  3_{31}, 1_{33}  
[3^{5,2,1}]  5_{21}, 2_{51}, 1_{52}  
[3,4,3,3]  16cell honeycomb 24cell honeycomb  
[6,3]  Hexagonal tiling and Triangular tiling  
[∞]  Apeirogon 
The group symbol 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.
There are infinitely many hyperbolic Coxeter groups describing reflection groups in hyperbolic space, notably including the hyperbolic triangle groups.
A Coxeter group is said to be irreducible if its Coxeter–Dynkin diagram is connected. Every Coxeter group is the direct product of the irreducible groups that correspond to the components of its Coxeter–Dynkin diagram.
A choice of reflection generators gives rise to a length function 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 (right) 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 diagrams 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.
Since a Coxeter group is generated by finitely many elements of order 2, its abelianization is an elementary abelian 2group, i.e., it is isomorphic to the direct sum of several copies of the cyclic group . This may be restated in terms of the first homology group of .
The Schur multiplier , equal to the second homology group of , was computed in (Ihara & Yokonuma 1965) for finite reflection groups and in (Yokonuma 1965) for affine reflection groups, with a more unified account given in (Howlett 1988). In all cases, the Schur multiplier is also an elementary abelian 2group. For each infinite family of finite or affine Weyl groups, the rank of stabilizes as goes to infinity.