Coxeter group

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

Symbol-free definition

A group is termed a Coxeter group if it can be equipped with a finite presentation given by the following relations:

• The square of every generator
• For some of the pairwise products of the generators, a certain power of that pairwise product

A presentation of this kind is termed a Coxeter presentation. Often, the term Coxeter group is used for the group along with a specific choice of Coxeter presentation.

Definition with symbols

A group $G$ is termed a Coxeter group if it can be equipped with a finite presentation with generators $s_i$ and relations:

• $s_i^2 = 1$
• $(s_is_j)^{m_{ij}} = 1$ where $m_{ij}$ is a symmetric function of $i$ and $j$ (for distinct $i$ and $j$)

Alternatively we can consider a symmetric matrix $m_{ij}$ with the diagonal entries being $1$ and simply require that for each $i$ and $j$ (not necessarily distinct) $(s_is_j)^{m_{ij}} = 1$. Note that we allow the entries $m_{ij}$ to be $0$.

Such a presentation is termed a Coxeter presentation and the matrix of $m_{ij}$s is termed a Coxeter matrix. Often, the term Coxeter group is used for a Coxeter group along with a specific choice of Coxeter presentation.

Particular cases

Number of generators	Form of Coxeter matrix	Common name for the group	Comment
2	$\begin{pmatrix} 1 & a \\ a & 1 \\\end{pmatrix}$	dihedral group of degree $a$, order $2a$.
2	$\begin{pmatrix} 1 & 2 \\ 2 & 1 \\\end{pmatrix}$	Klein four-group
2	$\begin{pmatrix} 1 & 3 \\ 3 & 1 \\\end{pmatrix}$	symmetric group of degree three
2	$\begin{pmatrix} 1 & 4 \\ 4 & 1 \\\end{pmatrix}$	dihedral group of order eight
2	$\begin{pmatrix} 1 & 5 \\ 5 & 1 \\\end{pmatrix}$	dihedral group of order ten
2	$\begin{pmatrix} 1 & 6 \\ 6 & 1 \\\end{pmatrix}$	dihedral group of order twelve
2	$\begin{pmatrix} 1 & 8 \\ 8 & 1 \\\end{pmatrix}$	dihedral group of order sixteen
3	$\begin{pmatrix} 1 & l & m \\ l & 1 & n \\ m & n & 1 \\\end{pmatrix}$	triangle group with parameters $(l,m,n)$
3	$\begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \\\end{pmatrix}$	elementary abelian group of order eight
3	$\begin{pmatrix} 1 & l & 2 \\ l & 1 & 2 \\ 2 & 2 & 1 \\\end{pmatrix}$	Direct product of dihedral group of degree $l$ (order $2l$) and cyclic group of order two
3	$\begin{pmatrix} 1 & 3 & 3 \\ 3 & 1 & 2 \\ 3 & 2 & 1 \\\end{pmatrix}$	symmetric group of degree four
3	$\begin{pmatrix} 1 & 4 & 3 \\ 4 & 1 & 2 \\ 3 & 2 & 1 \\\end{pmatrix}$	direct product of S4 and Z2
3	$\begin{pmatrix} 1 & 5 & 3 \\ 5 & 1 & 3 \\ 3 & 2 & 1 \\\end{pmatrix}$	direct product of A5 and Z2
3	$\begin{pmatrix} 1 & 7 & 3 \\ 7 & 1 & 2 \\ 3 & 2 & 1 \\\end{pmatrix}$	(7,3,2)-triangle group	this group is infinite.
$n$	1s on diagonal, 3s on superdiagonal and subdiagonal, 2s elsewhere.	symmetric group of degree $n + 1$.

Metaproperties

Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties

A direct product of Coxeter groups is a Coxeter group. The Coxeter matrix for the direct product is obtained by taking the block concatenation of the Coxeter matrices for the individual groups and then replacing the off-diagonal zero blocks by blocks with all entries equal to 2.