Coxeter group


In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups 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 groups, and finite Coxeter groups were classified in 1935.
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 infinite-dimensional Kac–Moody algebras.
Standard references include and.

Definition

Formally, a Coxeter group can be defined as a group with the presentation
where and for.
The condition means no relation of the form 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.
A number of conclusions can be drawn immediately from the above definition.
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 connected 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, affine type, or indefinite type. The indefinite type is sometimes further subdivided, e.g. into hyperbolic and other Coxeter groups. However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups.
Coxeter groupA1×A1A2B2H2G2A3B3D4
Coxeter diagram
Coxeter matrix
Schläfli matrix

An example

The graph in which vertices 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 Sn+1; the generators correspond to the transpositions,,...,. Two non-consecutive transpositions always commute, while gives the 3-cycle. Of course, this only shows that Sn+1 is a quotient group 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 groups. Simply put, Coxeter groups are abstract groups, while reflection groups are concrete groups. 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, while a Coxeter group is an abstract group generated by involutions, and whose relations have a certain form.
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, 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 groups of finite-dimensional Euclidean spaces.
The finite Coxeter groups consist of three one-parameter families of increasing rank one one-parameter family of dimension two, and six exceptional groups: and

Weyl groups

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 non-Weyl groups are the exceptions and and the family except where this coincides with one of the Weyl groups.
This can be proven by comparing the restrictions on 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 does not fill space; for the 120-cell does not fill space; for a p-gon does not tile the plane except for or .
Note further that the Dynkin diagrams Bn and Cn give rise to the same Weyl 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 cross-polytope being different regular polytopes but having the same symmetry group.

Properties

Some properties of the finite irreducible Coxeter groups are given in the following table. The order of reducible groups can be computed by the product of their irreducible subgroup orders.
Rank
n		Group
symbol		Alternate
symbol		Bracket
notation		Coxeter
graph		Reflections
m = ½nh		Coxeter number
h		OrderGroup structureRelated polytopes
1A1A1122
2A2A2336equilateral triangle|
3A3A36424regular tetrahedron|
4A4A41051205-cell|
5A5A51567205-simplex|
nAnAn...n/2n + 1!n-simplex
2B2C2448square|
3B3C39648cube| / regular octahedron|
4B4C4168384-tesseract| / 16-cell|
5B5C5251038405-cube| / 5-orthoplex|
nBnCn...n22n2n n!n-cube / n-orthoplex
4D4B4126192h / 16-cell|
5D5B52081920h / 5-orthoplex|
nDnBn...n22n−1 n!n-demicube / n-orthoplex
6E6E6361251840
221, 122
7E7E763182903040 321, 231, 132
8E8E812030696729600 421, 241, 142
4F4F42412115224-cell|
2G2– 6612hexagon|
2H2G25510pentagon|
3H3G31510120icosahedron| / dodecahedron|
4H4G4603014400120-cell| / 600-cell|
2I2Dnn2n
when n = pk + 1, p prime
when n = pk − 1, p prime		regular polygon|

Symmetry groups of regular polytopes

All symmetry groups of regular polytopes 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 n-simplex is the symmetric group Sn+1, also known as the Coxeter group of type An. The symmetry group of the n-cube and its dual, the n-cross-polytope, is Bn, 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 I2. In three dimensions, the symmetry group of the regular dodecahedron and its dual, the regular icosahedron, is H3, known as the full icosahedral group. In four dimensions, there are three special regular polytopes, the 24-cell, the 120-cell, and the 600-cell. The first has symmetry group F4, while the other two are dual and have symmetry group H4.
The Coxeter groups of type Dn, E6, E7, and E8 are the symmetry groups of certain semiregular polytopes.

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 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 An in this way, and the corresponding Coxeter group is the affine Weyl group of An. 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 is then the group generated by the reflections about all the hyperplanes in the diagram. 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 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.
A list of the affine Coxeter groups follows:
Group
symbol		Witt
symbol		Bracket notationCoxeter
graph		Related uniform tessellation
]...
or
...		Simplectic honeycomb
...Demihypercubic honeycomb
...Hypercubic honeycomb
...Demihypercubic honeycomb
or 222
or 331, 133
521, 251, 152
16-cell honeycomb
24-cell honeycomb
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.

Hyperbolic Coxeter groups

There are infinitely many hyperbolic Coxeter groups describing reflection groups in hyperbolic space, notably including the hyperbolic triangle groups.

Partial orders

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 generators is a reduced word. For example, the permutation in S3 has two reduced words, and. 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 weak order, the absolute order and the Bruhat order. An element v exceeds an element u in the Bruhat order if some reduced word for v contains a reduced word for u as a substring, where some letters 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 in S3 has only one reduced word,, so covers and in the Bruhat order but only covers in the weak order.

Homology

Since a Coxeter group is generated by finitely many elements of order 2, its abelianization is an elementary abelian 2-group, 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 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 2-group. For each infinite family of finite or affine Weyl groups, the rank of stabilizes as goes to infinity.