Coxeter group

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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 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 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.

Contents 1 Definition 2 An example 3 Finite Coxeter groups 3.1 Symmetry groups of regular polytopes 4 Affine Weyl groups 5 Hyperbolic Coxeter groups 6 Partial orders 7 References 8 See also 9 External links

[edit] Definition

Formally, a Coxeter group can be defined as a group with the presentation

where m_ii = 1 and m_ij ≥ 2 for i ≠ j. The condition m_ij = ∞ means no relation of the form (r_i r_j)^m should be imposed.

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

• The relation m_ii = 1 means that (r_i)² = 1 for all i ; the generators are involutions.
• If m_ij = 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.

• In order to avoid redundancy among the relations, it is necessary to assume that m_ij=m_ji. This follows by observing that

yy = 1,

together with

(xy)^m = 1

implies that

(yx)^m = (yx)^myy = y(xy)^my = yy = 1.

The Coxeter matrix is the n×n, symmetric matrix with entries m_ij. Indeed, every symmetric matrix with positive integer and ∞ entries and with 1's on the diagonal 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_ij ≥ 3.
• An edge is labelled with the value of m_ij 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 components, the associated group is the direct product of the groups associated to the individual components.

[edit] 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 S_n+1; the generators correspond to the transpositions (1 2), (2 3), ... (n n+1). Two non-consecutive transpositions always commute, while (k k+1) (k+1 k+2) gives the 3-cycle (k k+2 k+1). Of course this only shows that S_n+1 is a quotient group of the Coxeter group, but it is not too difficult to check that equality holds.

[edit] Finite Coxeter groups

Every Weyl group can be realized as a Coxeter group. The Coxeter graph can be obtained from the Dynkin diagram by replacing every double edge with an edge labelled 4 and every triple edge by an edge labelled 6. The example given above corresponds to the Weyl group of the root system of type A_n. The Weyl groups include most of the finite Coxeter groups, but there are additional examples as well. The following list gives all connected Coxeter graphs giving rise to finite groups:

Comparing this with the list of simple root systems, we see that B_n and C_n give rise to the same Coxeter group. Also, G₂ appears to be missing, but it is present under the name I₂(6). The additions to the list are H₃, H₄, and the I₂(p).

Some properties of the finite Coxeter groups are given in the following table:

Group symbol	Alternate symbol	Rank	Order	Related polytopes	Coxeter-Dynkin diagram
An	An	n	(n + 1)!	n-simplex	...
Bn = Cn	Cn	n	2n n!	n-hypercube / n-cross-polytope	...
Dn	Bn	n	2n−1 n!	demihypercube	...
I2(p)	D2p	2	2p	p-gon
H3	G3	3	120	icosahedron / dodecahedron
F4	F4	4	1152	24-cell
H4	G4	4	14400	120-cell / 600-cell
E6	E6	6	51840	E6 polytope
E7	E7	7	2903040	E7 polytope
E8	E8	8	696729600	E8 polytope

[edit] Symmetry groups of regular polytopes

All symmetry groups of regular polytopes are finite Coxeter groups. The dihedral groups, which are the symmetry groups of regular polygons, form the series I₂(p). The symmetry group of a regular n-simplex is the symmetric group S_n+1, also known as the Coxeter group of type A_n. The symmetry group of the n-cube is the same as that of the n-cross-polytope, namely BC_n. The symmetry group of the regular dodecahedron and the regular icosahedron is H₃. In dimension 4, 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 have symmetry group H₄.

The Coxeter groups of type D_n, E₆, E₇, and E₈ are the symmetry groups of certain semiregular polytopes.

[edit] Affine Weyl groups

The affine Weyl 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 Weyl group, and the Coxeter graph is obtained from the Coxeter graph of the Weyl 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	Alternate symbol	Related uniform tessellation(s)	Coxeter-Dynkin diagram
A~n-1	Pn	Simplex-rectified-simplex honeycomb A~2:Triangular tiling A~3:Tetrahedral-octahedral honeycomb	...
B~n-1	Rn	Hypercubic honeycomb	...
C~n-1	Sn	Demihypercubic honeycomb	...
D~n-1	Qn	Demihypercubic honeycomb	...
I~1	W2	apeirogon
H~2	G3	Hexagonal tiling and Triangular tiling
F~4	V5	Hexadecachoric tetracomb and Icositetrachoronic tetracomb or F4 lattice
E~6	T7	E6 lattice
E~7	T8	E7 lattice
E~8	T9	E8 lattice

Note 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.

[edit] Hyperbolic Coxeter groups

There are also hyperbolic Coxeter groups describing reflection groups in hyperbolic space, though they won't act cocompactly in dimensions larger than 29.

[edit] 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. An expression for v using l(v) generators is a reduced word. For example, the permutation (13) in S₃ has two reduced words, (12)(23)(12) and (23)(12)(23).

Using reduced words one may define two partial orders on the Coxeter group, the weak order and the Bruhat order. 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.

For example, the permutation (1 2 3) in S₃ has only one reduced word, (12)(23), so covers (12) and (23) in the Bruhat order but only covers (12) in the weak order.

[edit] References

• 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 Björner and Francesco Brenti, Combinatorics of Coxeter Groups, Graduate texts in mathematics, vol. 231, Springer, (2005)
• E. Vinberg, Absence of crystallographic groups of reflections in Lobachevski spaces of large dimension, Trudy Moskov. Mat. Obshch. 47 (1984)

[edit] See also

• Artin group
• Weyl group
• Triangle group
• Coxeter number
• Complex reflection group
• Chevalley–Shephard–Todd theorem
• Kazhdan-Lusztig polynomial
• Field with one element

[edit] External links

• Eric W. Weisstein, Coxeter group at MathWorld.
• Jenn, software for visualizing the Cayley graphs of finite Coxeter groups on up to four generators.