Coxeter group

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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:

• $s_i^2=1$
• $(s_i{s}_j{)}^{m_{ij}}=1$ 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) $(s_i{s}_j{)}^{m_{ij}}=1$. 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	$\left(\begin{array}{cc}1& a\\ a& 1\\ \end{array}\right)$	dihedral group of degree $a$, order $2a$.
2	$\left(\begin{array}{cc}1& 2\\ 2& 1\\ \end{array}\right)$	Klein four-group
2	$\left(\begin{array}{cc}1& 3\\ 3& 1\\ \end{array}\right)$	symmetric group of degree three
2	$\left(\begin{array}{cc}1& 4\\ 4& 1\\ \end{array}\right)$	dihedral group of order eight
2	$\left(\begin{array}{cc}1& 5\\ 5& 1\\ \end{array}\right)$	dihedral group of order ten
2	$\left(\begin{array}{cc}1& 6\\ 6& 1\\ \end{array}\right)$	dihedral group of order twelve
2	$\left(\begin{array}{cc}1& 8\\ 8& 1\\ \end{array}\right)$	dihedral group of order sixteen
3	$\left(\begin{array}{ccc}1& l& m\\ l& 1& n\\ m& n& 1\\ \end{array}\right)$	triangle group with parameters $(l,m,n)$
3	$\left(\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\\ \end{array}\right)$	elementary abelian group of order eight
3	$\left(\begin{array}{ccc}1& l& 2\\ l& 1& 2\\ 2& 2& 1\\ \end{array}\right)$	Direct product of dihedral group of degree $l$ (order $2l$) and cyclic group of order two
3	$\left(\begin{array}{ccc}1& 3& 3\\ 3& 1& 2\\ 3& 2& 1\\ \end{array}\right)$	symmetric group of degree four
3	$\left(\begin{array}{ccc}1& 4& 3\\ 4& 1& 2\\ 3& 2& 1\\ \end{array}\right)$	direct product of S4 and Z2
3	$\left(\begin{array}{ccc}1& 5& 3\\ 5& 1& 3\\ 3& 2& 1\\ \end{array}\right)$	direct product of A5 and Z2
3	$\left(\begin{array}{ccc}1& 7& 3\\ 7& 1& 2\\ 3& 2& 1\\ \end{array}\right)$	(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.