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; 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, 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.

Definition

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.
A number of conclusions can be drawn immediately from the above definition.

The relation means that for all ; as such the generators are involutions.
If, then the generators and commute. This follows by observing that

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.

Coxeter matrix and Schläfli matrix

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.

The vertices of the graph are labelled by generator subscripts.
Vertices and are adjacent if and only if.
An edge is labelled with the value of whenever the value is or greater.

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, 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 group	A₁×A₁	A₂	B₂	I₂	G₂	A₃	B₃	D₄
Coxeter diagram
Coxeter matrix
Schläfli matrix

An example

The graph in which vertices through 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 ; the generators correspond to the transpositions. Any two non-consecutive transpositions commute, while multiplying two consecutive transpositions gives a 3-cycle :. Therefore is a quotient of the Coxeter group having Coxeter diagram. Further arguments show that this quotient map is an isomorphism.

Abstraction of reflection groups

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 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. The converse is partially true: every finite Coxeter group admits a faithful representation as a finite reflection group of some Euclidean space.
However, not every infinite Coxeter group admits a representation as a reflection group.
Finite Coxeter groups have been classified.

Finite Coxeter groups

Classification

Finite Coxeter groups are classified in terms of their Coxeter diagrams.
The finite Coxeter groups with connected Coxeter diagrams consist of three one-parameter families of increasing dimension, a one-parameter family of dimension two, and six exceptional groups. Every finite Coxeter group is the direct product of finitely many of these irreducible groups.

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 ones are the exceptions and and those members of the family that are not exceptionally isomorphic to a Weyl group.
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 B_n and C_n 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 a reducible group can be computed by the product of its irreducible subgroup orders.

			Bracket notation	Coxeter graph	Reflections	Coxeter number h	Order	Group structure	Related polytopes
1	A₁	A₁			1	2	2		Line segment\|
2	A₂	A₂			3	3	6		equilateral triangle\|
3	A₃	A₃			6	4	24		regular tetrahedron\|
4	A₄	A₄			10	5	120		5-cell\|
5	A₅	A₅			15	6	720		5-simplex\|
n	A_n	A_n		...	n/2	n + 1	!		n-simplex
2	B₂	C₂			4	4	8		square\|
3	B₃	C₃			9	6	48		cube\| / regular octahedron\|
4	B₄	C₄			16	8	384		tesseract\| / 16-cell\|
5	B₅	C₅			25	10	3840		5-cube\| / 5-orthoplex\|
n	B_n	C_n		...	n²	2n	2ⁿ n!		n-cube / n-orthoplex
4	D₄	B₄			12	6	192		h / 16-cell\|
5	D₅	B₅			20	8	1920		h / 5-orthoplex\|
n	D_n	B_n		...	n	2	2ⁿ⁻¹ n!		n-demicube / n-orthoplex
6	E₆	E₆			36	12	51840		2₂₁, 1₂₂
7	E₇	E₇			63	18	2903040		3₂₁, 2₃₁, 1₃₂
8	E₈	E₈			120	30	696729600		4₂₁, 2₄₁, 1₄₂
4	F₄	F₄			24	12	1152		24-cell\|
2	G₂	–			6	6	12		hexagon\|
2	I₂	G₂			5	5	10		pentagon\|
3	H₃	G₃			15	10	120		icosahedron\| / dodecahedron\|
4	H₄	G₄			60	30	14400		120-cell\| / 600-cell\|
2	I₂	D			n	n	2n	when n = p^k + 1, p prime when n = p^k − 1, p prime	regular polygon\|