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 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

Definition

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

\left\langle r_1,r_2,\ldots,r_n \mid (r_ir_j)^{m_{ij}}=1\right\rangle

where mi i = 1 and mi j ≥ 2 for i ≠ j. The condition mi j = ∞ means no relation of the form (ri rj )m should be imposed.

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

xx = yy = 1,
together with
xyxy = 1
implies that
xy = x(xyxy)y = (xx)yx(yy) = yx.
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 mi j. 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.

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.

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 (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 Sn+1 is a quotient group of the Coxeter group, but it is not too difficult to check that equality holds.

Finite Coxeter groups

Coxeter graphs of the 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 An. 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 Bn and Cn 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 CDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.png...CDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
Bn = Cn Cn n 2n n! n-hypercube / n-cross-polytope CDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.png...CDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
Dn Bn n 2n−1 n! demihypercube CD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.png...CD 3b.pngCD dot.pngCD 3b.pngCD dot.png
I2(p) D2p 2 2p p-gon CDW dot.pngCDW p.pngCDW dot.png
H3 G3 3 120 icosahedron / dodecahedron CDW dot.pngCDW 5.pngCDW dot.pngCDW 3b.pngCDW dot.png
F4 F4 4 1152 24-cell CDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.png
H4 G4 4 14400 120-cell / 600-cell CDW dot.pngCDW 5.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
E6 E6 6 51840 221 polytope CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.png
E7 E7 7 2903040 321 polytope CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.png
E8 E8 8 696729600 421 polytope CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.png

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 Sn+1, also known as the Coxeter group of type An. The symmetry group of the n-cube is the same as that of the n-cross-polytope, namely BCn. 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 Dn, E6, E7, and E8 are the symmetry groups of certain semiregular polytopes.

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 An in this way, and the corresponding Coxeter group is the affine Weyl group of An. 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:

Affine coxeter.PNG
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		 CD downbranch-00.pngCD downbranch-33.pngCD downbranch-open.pngCD downbranch-33.png...CD downbranch-33.pngCD downbranch-open.pngCD downbranch-33.pngCD downbranch-00.png
B~n-1 Rn Hypercubic honeycomb CDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.png...CDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.png
C~n-1 Sn Demihypercubic honeycomb CD dot.pngCD 4.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.png...CD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.png
D~n-1 Qn Demihypercubic honeycomb CD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.png...CD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.png
I~1 W2 apeirogon CDW dot.pngCDW infin.pngCDW dot.png
H~2 G3 Hexagonal tiling and
Triangular tiling		 CDW dot.pngCDW 6.pngCDW dot.pngCDW 3.pngCDW dot.png
F~4 V5 Hexadecachoric honeycomb and
Icositetrachoric honeycomb or
F4 lattice		 CDW dot.pngCDW 3b.pngCDW dot.pngCDW 4.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
E~6 T7 E6 lattice CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD downbranch-33.pngCD downbranch-open.pngCD 3b.pngCD dot.png
E~7 T8 E7 lattice CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.png
E~8 T9 E8 lattice CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.png

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.

Hyperbolic Coxeter groups

There are also hyperbolic Coxeter groups describing reflection groups in hyperbolic space.

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.

References

See also

External links