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Coxeter groups

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Coxeter groups

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 (Coxeter 1934) as abstractions of reflection groups, and finite Coxeter groups were classified in 1935 (Coxeter 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 (Humphreys 1990) and (Davis 2007).

Definition

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

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

where $m_\left\{ii\right\}=1$ and $m_\left\{ij\right\}\geq 2$ for $i\neq j$. The condition $m_\left\{i j\right\}= \infty$ means no relation of the form $\left(r_i r_j\right)^m$ should be imposed.

The pair (W,S) where W is a Coxeter group with generators S={r1,...,rn} is called Coxeter system. Note that in general S is not uniquely determined by W. For example, the Coxeter groups of type BC3 and A1xA3 are isomorphic but the Coxeter systems are not equivalent (see below for an explanation of this notation).

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

• The relation mi i = 1 means that (riri )1 = (ri )2 = 1 for all i ; the generators are involutions.
• If mi j = 2, then the generators ri and rj commute. This follows by observing that
xx = yy = 1,
together with
xyxy = 1
implies that
xy = x(xyxy)y = (xx)yx(yy) = yx.
Alternatively, since the generators are involutions, $r_i = r_i^\left\{-1\right\}$, so $\left(r_ir_j\right)^2=r_ir_jr_ir_j=r_ir_jr_i^\left\{-1\right\}r_j^\left\{-1\right\}$, and thus is equal to the commutator.
• In order to avoid redundancy among the relations, it is necessary to assume that mi j = mj i. This follows by observing that
yy = 1,
together with
(xy)m = 1
implies that
(yx)m = (yx)myy = y(xy)my = yy = 1.
Alternatively, $\left(xy\right)^m$ and $\left(yx\right)^m$ are conjugate elements, as $y\left(xy\right)^m y^\left\{-1\right\} = \left(yx\right)^m yy^\left\{-1\right\}=\left(yx\right)^m$.

Coxeter matrix and Schläfli matrix

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 such that all nondiagonal entries are greater than 1 serves to define a Coxeter group.

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 i and j are connected if and only if mi j ≥ 3.
• An edge is labelled with the value of mi j 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. Thus the disjoint union of Coxeter graphs yields a direct product of Coxeter groups.

The Coxeter matrix, Mi,j, is related to the Schläfli matrix, Ci,j, but the elements are modified, being proportional to the dot product of the pairwise generators: Schläfli matrix Ci,j=-2cos(π/Mi,j). The Schläfli matrix is useful because its eigenvalues determine whether the Coxeter group is of finite type (all positive), affine type (all non-negative, 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 non-equivalent definitions for hyperbolic Coxeter groups.

Examples
Coxeter group A1×A1 A2 $\left\{\tilde\left\{I\right\}\right\}_1$ A3 BC3 D4 $\left\{\tilde\left\{A\right\}\right\}_3$
Coxeter diagram Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
Coxeter matrix $\left \left[$

\begin{smallmatrix}

1 &  2 \\
2 &  1  \\

\end{smallmatrix}\right ]

$\left \left[$

\begin{smallmatrix}

1 &  3 \\
3 &  1  \\

\end{smallmatrix}\right ]

$\left \left[$

\begin{smallmatrix}

1 &  \infty \\
\infty &  1  \\

\end{smallmatrix}\right ]

$\left \left[$

\begin{smallmatrix}

1 &  3 &  2 \\
3 &  1 &  3 \\
2 &  3 &  1

\end{smallmatrix}\right ]

$\left \left[$

\begin{smallmatrix}

1 &  4 &  2 \\
4 &  1 &  3 \\
2 &  3 &  1

\end{smallmatrix}\right ]

$\left \left[$

\begin{smallmatrix}

1 &  3 &  2 & 2 \\
3 &  1 &  3 & 3 \\
2 &  3 &  1 & 2\\
2 &  3 &  2 & 1

\end{smallmatrix}\right ]

$\left \left[$

\begin{smallmatrix}

1 &  3 &  2 & 3 \\
3 &  1 &  3 & 2 \\
2 &  3 &  1 & 3\\
3 &  2 &  3 & 1

\end{smallmatrix}\right ]

Schläfli matrix $\left \left[$

\begin{smallmatrix}

2 &  0 \\
0 &  2

\end{smallmatrix}\right ]

$\left \left[$

\begin{smallmatrix}

2 &  -1 \\
-1 &  2

\end{smallmatrix}\right ]

$\left \left[$

\begin{smallmatrix}

2 &  -2 \\
-2 &  2

\end{smallmatrix}\right ]

$\left \left[$

\begin{smallmatrix}

2 &  -1 &  0 \\
-1 &  2 &  -1 \\
0 &  -1 &  2

\end{smallmatrix}\right ]

$\left \left[$

\begin{smallmatrix}

2 & -\sqrt{2} &  0 \\
-\sqrt{2} &  2 &  -1 \\
0 &  -1 &  2

\end{smallmatrix}\right ]

$\left \left[$

\begin{smallmatrix}

2 & -1  & 0 & 0 \\

-1 & 2 & -1 & -1 \\

0 & -1 &  2 & 0\\
0 & -1 &  0 & 2

\end{smallmatrix}\right ]

$\left \left[$

\begin{smallmatrix}

2 & -1  & 0 & -1 \\

-1 & 2 & -1 & 0 \\

0 & -1 &  2 & -1\\

-1 & 0 & -1 & 2 \end{smallmatrix}\right ]

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 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 (given via a presentation), while reflection groups are concrete groups (given as subgroups of linear groups or various generalizations). 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 (which have order 2), while a Coxeter group is an abstract group generated by involutions (elements of order 2, abstracting from reflections), and whose relations have a certain form ($\left(r_ir_j\right)^k$, corresponding to hyperplanes meeting at an angle of $\pi/k$, with $r_ir_j$ being of order k abstracting from a rotation by $2\pi/k$).

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, (Coxeter 1934) proved that every reflection group is a Coxeter group (i.e., has a presentation where all relations are of the form $r_i^2$ or $\left(r_ir_j\right)^k$), and indeed this paper introduced the notion of a Coxeter group, while (Coxeter 1935) 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 (Coxeter 1935), 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 $A_n, BC_n, D_n,$ one one-parameter family of dimension two, $I_2\left(p\right),$ and six exceptional groups: $E_6, E_7, E_8, F_4, H_3,$ and $H_4.$

Weyl groups

Main article: Weyl group

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 $A_n, BC_n,$ and $D_n,$ and the exceptions $E_6, E_7, E_8, F_4,$ and $I_2\left(6\right),$ denoted in Weyl group notation as $G_2.$ The non-Weyl groups are the exceptions $H_3$ and $H_4,$ and the family $I_2\left(p\right)$ except where this coincides with one of the Weyl groups (namely $I_2\left(3\right) \cong A_2, I_2\left(4\right) \cong BC_2,$ and $I_2\left(6\right) \cong G_2$).

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.[1] 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 $H_3,$ the dodecahedron (dually, icosahedron) does not fill space; for $H_4,$ the 120-cell (dually, 600-cell) does not fill space; for $I_2\left(p\right)$ a p-gon does not tile the plane except for $p=3, 4,$ or $6$ (the triangular, square, and hexagonal tilings, respectively).

Note further that the (directed) Dynkin diagrams Bn and Cn 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 cross-polytope being different regular polytopes but having the same symmetry group.

Properties

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

Group
symbol
Alternate
symbol
Bracket notation Rank Order Related polytopes Coxeter-Dynkin diagram
An An [3n-1] n (n + 1)! n-simplex Template:CDD..Template:CDD
BCn Cn [4,3n-2] n 2n n! n-hypercube / n-cross-polytope Template:CDD...Template:CDD
Dn Bn [3n-3,1,1] n 2n−1 n! n-demihypercube Template:CDD...Template:CDD
E6 E6 [32,2,1] 6 72x6! = 51840 221, 122 Template:CDD or Template:CDD
E7 E7 [33,2,1] 7 72x8! = 2903040 321, 231, 132 Template:CDD
E8 E8 [34,2,1] 8 192x10! = 696729600 421, 241, 142 Template:CDD
F4 F4 [3,4,3] 4 1152 24-cell Template:CDD
G2 - [6] 2 12 hexagon Template:CDD
H2 G2 [5] 2 10 pentagon Template:CDD
H3 G3 [3,5] 3 120 icosahedron / dodecahedron Template:CDD
H4 G4 [3,3,5] 4 14400 120-cell / 600-cell Template:CDD
I2(p) D2p [p] 2 2p p-gon Template:CDD

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 BCn, 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(p). 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.

Table of polytope families
Group An BCn Dn
 E6 E7 E8 F4 G2
Hn In(p)
2
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Template:CDD

(Triangle)

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Template:CDD

Square

Hexagon Pentagon Template:CDD
p-gon
3
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Tetrahedron
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Template:CDD
Cube
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Template:CDD
Octahedron
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Template:CDD
Tetrahedron
Dodecahedron Icosahedron
4
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5-cell
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16-cell
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24-cell
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120-cell
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600-cell
5
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5-simplex
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5-cube
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5-orthoplex
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5-demicube

6
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6-simplex
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6-cube
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6-orthoplex
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6-demicube

7
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7-simplex
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7-cube
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7-orthoplex
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7-demicube

8
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8-simplex
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8-cube
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8-orthoplex
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8-demicube
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241

9
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9-simplex
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9-cube
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9-orthoplex
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9-demicube

10
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10-simplex
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10-cube
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10-orthoplex
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10-demicube

Family
n
n-simplex n-hypercube n-orthoplex n-demicube 1k2 2k1 k21

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 is obtained from the Coxeter graph of the Coxeter 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:

Group
symbol
Witt
symbol
Bracket notation Related uniform tessellation(s) Coxeter-Dynkin diagram
$\left\{\tilde\left\{A\right\}\right\}_n$ Pn+1 [3[n]] Simplectic honeycomb Template:CDD...Template:CDD
or
Template:CDD...Template:CDD
$\left\{\tilde\left\{B\right\}\right\}_n$ Sn+1 [4,3n-3,31,1] Demihypercubic honeycomb Template:CDD...Template:CDD
$\left\{\tilde\left\{C\right\}\right\}_n$ Rn+1 [4,3n-2,4] Hypercubic honeycomb Template:CDD...Template:CDD
$\left\{\tilde\left\{D\right\}\right\}_n$ Qn+1 [ 31,1,3n-4,31,1] Demihypercubic honeycomb Template:CDD...Template:CDD
$\left\{\tilde\left\{E\right\}\right\}_6$ T7 [32,2,2] 222 Template:CDD or Template:CDD
$\left\{\tilde\left\{E\right\}\right\}_7$ T8 [33,3,1] 331, 133 Template:CDD or Template:CDD
$\left\{\tilde\left\{E\right\}\right\}_8$ T9 [35,2,1] 521, 251, 152 Template:CDD
$\left\{\tilde\left\{F\right\}\right\}_4$ U5 [3,4,3,3] 16-cell honeycomb
24-cell honeycomb
Template:CDD
$\left\{\tilde\left\{G\right\}\right\}_2$ V3 [6,3] Hexagonal tiling and
Triangular tiling
Template:CDD
$\left\{\tilde\left\{I\right\}\right\}_1$ W2 [∞] apeirogon Template:CDD

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 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(v) generators is a reduced word. For example, the permutation (13) in S3 has two reduced words, (12)(23)(12) and (23)(12)(23). The function $v \to \left(-1\right)^\left\{l\left(v\right)\right\}$ defines a map $G \to \\left\{\pm 1\\right\},$ 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 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.

Homology

Since a Coxeter group W 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 Z2. This may be restated in terms of the first homology group of W.

The Schur multiplier M(W) (related to the second homology) 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 2-group. For each infinite family {Wn} of finite or affine Weyl groups, the rank of M(W) stabilizes as n goes to infinity.

References

• =1

|journal=J. London Math. Soc. |volume=10 |year=1935 |pages=21–25 }}

• 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)
• Howard Hiller, Geometry of Coxeter groups. Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. iv+213 pp. ISBN 0-273-08517-4
• Nicolas Bourbaki, Lie Groups and Lie Algebras: Chapter 4-6, Elements of Mathematics, Springer (2002). ISBN 978-3-540-42650-9