Coxeter group

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

${\displaystyle \left\langle r_{1},r_{2},\ldots ,r_{n}\mid (r_{i}r_{j})^{m_{ij}}=1\right\rangle }$

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

The pair ${\displaystyle (W,S)}$ where ${\displaystyle W}$ is a Coxeter group with generators ${\displaystyle S=\{r_{1},\dots ,r_{n}\}}$ is called a Coxeter system. Note that in general ${\displaystyle S}$ is not uniquely determined by ${\displaystyle W}$. For example, the Coxeter groups of type ${\displaystyle B_{3}}$ and ${\displaystyle A_{1}\times A_{3}}$ 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 ${\displaystyle m_{ii}=1}$ means that ${\displaystyle (r_{i}r_{i})^{1}=(r_{i})^{2}=1}$ for all ${\displaystyle i}$ ; as such the generators are involutions.
• If ${\displaystyle m_{ij}=2}$, then the generators ${\displaystyle r_{i}}$ and ${\displaystyle r_{j}}$ commute. This follows by observing that

${\displaystyle xx=yy=1}$,

together with

${\displaystyle xyxy=1}$

implies that

${\displaystyle xy=x(xyxy)y=(xx)yx(yy)=yx}$.

Alternatively, since the generators are involutions, ${\displaystyle r_{i}=r_{i}^{-1}}$, so ${\displaystyle (r_{i}r_{j})^{2}=r_{i}r_{j}r_{i}r_{j}=r_{i}r_{j}r_{i}^{-1}r_{j}^{-1}}$, and thus is equal to the commutator.

• In order to avoid redundancy among the relations, it is necessary to assume that ${\displaystyle m_{ij}=m_{ji}}$. This follows by observing that

${\displaystyle yy=1}$,

together with

${\displaystyle (xy)^{m}=1}$

implies that

${\displaystyle (yx)^{m}=(yx)^{m}yy=y(xy)^{m}y=yy=1}$.

Alternatively, ${\displaystyle (xy)^{k}}$ and ${\displaystyle (yx)^{k}}$ are conjugate elements, as ${\displaystyle y(xy)^{k}y^{-1}=(yx)^{k}yy^{-1}=(yx)^{k}}$.

Coxeter matrix and Schläfli matrix

The Coxeter matrix is the ${\displaystyle n\times n}$, symmetric matrix with entries ${\displaystyle m_{ij}}$. Indeed, every symmetric matrix with diagonal entries exclusively 1 and nondiagonal entries in the set ${\displaystyle \{2,3,\ldots \}\cup \{\infty \}}$ 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 ${\displaystyle i}$ and ${\displaystyle j}$ are adjacent if and only if ${\displaystyle m_{ij}\geq 3}$.
• An edge is labelled with the value of ${\displaystyle m_{ij}}$ whenever the value is ${\displaystyle 4}$ 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, ${\displaystyle M_{ij}}$, is related to the ${\displaystyle n\times n}$ Schläfli matrix ${\displaystyle C}$ with entries ${\displaystyle C_{ij}=-2\cos(\pi /M_{ij})}$, 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 (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	B2	H2	G2	${\displaystyle {\tilde {I}}_{1}}$	A3	B3	D4	${\displaystyle {\tilde {A}}_{3}}$
Coxeter diagram
Coxeter matrix	${\displaystyle \left[{\begin{smallmatrix}1&2\\2&1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&3\\3&1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&4\\4&1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&5\\5&1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&6\\6&1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&\infty \\\infty &1\\\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&3&2\\3&1&3\\2&3&1\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&4&2\\4&1&3\\2&3&1\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&3&2&2\\3&1&3&3\\2&3&1&2\\2&3&2&1\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&3&2&3\\3&1&3&2\\2&3&1&3\\3&2&3&1\end{smallmatrix}}\right]}$
Schläfli matrix	${\displaystyle \left[{\begin{smallmatrix}2&0\\0&2\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}\ \,2&-1\\-1&\ \,2\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}\ \,2&-{\sqrt {2}}\\-{\sqrt {2}}&\ \,2\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}\ \,2&-\phi \\-\phi &\ \,2\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}\ \,2&-{\sqrt {3}}\\-{\sqrt {3}}&\ \,2\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}\ \,2&-2\\-2&\ \,2\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}\ \,2&-1&\ \,0\\-1&\ \,2&-1\\\ \,0&-1&\ \,2\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}\ \,\ \ 2&-{\sqrt {2}}&\ \,0\\-{\sqrt {2}}&\ \,\ \ 2&-1\\\ \,\ \ 0&\ \,-1&\ \,2\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}\ \,2&-1&\ \,0&\ \,0\\-1&\ \,2&-1&-1\\\ \,0&-1&\ \,2&\ \,0\\\ \,0&-1&\ \,0&\ \,2\end{smallmatrix}}\right]}$	${\displaystyle \left[{\begin{smallmatrix}\ \,2&-1&\ \,0&-1\\-1&\ \,2&-1&\ \,0\\\ \,0&-1&\ \,2&-1\\-1&\ \,0&-1&\ \,2\end{smallmatrix}}\right]}$

The graph ${\displaystyle A_{n}}$ 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 described by the graph, but it is not too difficult to check that equality holds.

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 (${\displaystyle (r_{i}r_{j})^{k}}$, corresponding to hyperplanes meeting at an angle of ${\displaystyle \pi /k}$, with ${\displaystyle r_{i}r_{j}}$ being of order k abstracting from a rotation by ${\displaystyle 2\pi /k}$).

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 ${\displaystyle r_{i}^{2}}$ or ${\displaystyle (r_{i}r_{j})^{k}}$), 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.

Coxeter diagrams for the Affine Coxeter groups

Stiefel diagram for the ${\displaystyle G_{2}}$ root system

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 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 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 (or affine Weyl group) is then the group generated by the (affine) reflections about all the hyperplanes in the diagram.[4] 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 ${\displaystyle G_{2}}$ root system.

Suppose ${\displaystyle R}$ is an irreducible root system of rank ${\displaystyle r>1}$ and let ${\displaystyle \alpha _{1},\ldots ,\alpha _{r}}$ be a collection of simple roots. Let, also, ${\displaystyle \alpha _{r+1}}$ denote the highest root. Then the affine Coxeter group is generated by the ordinary (linear) reflections about the hyperplanes perpendicular to ${\displaystyle \alpha _{1},\ldots ,\alpha _{r}}$, together with an affine reflection about a translate of the hyperplane perpendicular to ${\displaystyle \alpha _{r+1}}$. The Coxeter graph for the affine Weyl group is the Coxeter–Dynkin diagram for ${\displaystyle R}$, together with one additional node associated to ${\displaystyle \alpha _{r+1}}$. 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 ${\displaystyle \alpha _{r+1}}$.[5]

A list of the affine Coxeter groups follows:

Group symbol	Witt symbol	Bracket notation	Coxeter graph	Related uniform tessellation(s)
${\displaystyle {\tilde {A}}_{n}}$	${\displaystyle P_{n+1}}$	[3[n]]	... or ...	Simplectic honeycomb
${\displaystyle {\tilde {B}}_{n}}$	${\displaystyle S_{n+1}}$	[4,3n − 3,31,1]	...	Demihypercubic honeycomb
${\displaystyle {\tilde {C}}_{n}}$	${\displaystyle R_{n+1}}$	[4,3n−2,4]	...	Hypercubic honeycomb
${\displaystyle {\tilde {D}}_{n}}$	${\displaystyle Q_{n+1}}$	[ 31,1,3n−4,31,1]	...	Demihypercubic honeycomb
${\displaystyle {\tilde {E}}_{6}}$	${\displaystyle T_{7}}$	[32,2,2]	or	222
${\displaystyle {\tilde {E}}_{7}}$	${\displaystyle T_{8}}$	[33,3,1]	or	331, 133
${\displaystyle {\tilde {E}}_{8}}$	${\displaystyle T_{9}}$	[35,2,1]		521, 251, 152
${\displaystyle {\tilde {F}}_{4}}$	${\displaystyle U_{5}}$	[3,4,3,3]		16-cell honeycomb 24-cell honeycomb
${\displaystyle {\tilde {G}}_{2}}$	${\displaystyle V_{3}}$	[6,3]		Hexagonal tiling and Triangular tiling
${\displaystyle {\tilde {I}}_{1}}$	${\displaystyle W_{2}}$	[∞]		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.

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

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 ${\displaystyle v\to (-1)^{\ell (v)}}$ defines a map ${\displaystyle G\to \{\pm 1\},}$ 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.

Since a Coxeter group ${\displaystyle W}$ 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 ${\displaystyle Z_{2}}$. This may be restated in terms of the first homology group of ${\displaystyle W}$.

The Schur multiplier ${\displaystyle M(W)}$, equal to the second homology group of ${\displaystyle W}$, 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 ${\displaystyle \{W_{n}\}}$ of finite or affine Weyl groups, the rank of ${\displaystyle M(W_{n})}$ stabilizes as ${\displaystyle n}$ goes to infinity.

• Artin–Tits group
• Chevalley–Shephard–Todd theorem
• Complex reflection group
• Coxeter element
• Iwahori–Hecke algebra, a quantum deformation of the group algebra
• Kazhdan–Lusztig polynomial
• Longest element of a Coxeter group
• Supersolvable arrangement

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• Jenn software for visualizing the Cayley graphs of finite Coxeter groups on up to four generators