Editing Coxeter group (section)

==Definition==
Formally, a '''Coxeter group''' can be defined as a group with the [[Presentation of a group|presentation]]

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

where <math>m_{ii}=1</math> and <math>m_{ij} = m_{ji} \ge 2</math> is either an integer or <math> \infty </math> for <math>i\neq j</math>.
Here, the condition <math>m_{i j}=\infty</math>  means that no relation of the form <math>(r_ir_j)^m = 1</math> for any integer <math> m \ge 2</math> should be imposed.

The pair <math>(W,S)</math> where <math>W</math> is a Coxeter group with generators <math>S=\{r_1, \dots , r_n\}</math> is called a '''Coxeter system'''. Note that in general <math>S</math> is ''not'' uniquely determined by <math>W</math>. For example, the Coxeter groups of type <math>B_3</math> and <math>A_1\times A_3</math> are isomorphic but the Coxeter systems are not equivalent, since the former has 3 generators and the latter has 1 + 3 = 4 generators (see below for an explanation of this notation).

A number of conclusions can be drawn immediately from the above definition.
* The relation <math>m_{ii} = 1</math> means that <math>(r_ir_i)^1 = (r_i)^2 = 1</math> for all <math>i</math>&nbsp;; as such the generators are [[involution (mathematics)|involution]]s.
* If <math>m_{ij} = 2</math>, then the generators <math>r_i</math> and <math>r_j</math> commute.  This follows by observing that
::<math>xx = yy = 1</math>,
: together with
::<math>xyxy = 1</math>
: implies that
::<math>xy = x(xyxy)y = (xx)yx(yy) = yx</math>.
:Alternatively, since the generators are involutions, <math>r_i = r_i^{-1}</math>, so <math>1 =(r_ir_j)^2=r_ir_jr_ir_j=r_ir_jr_i^{-1}r_j^{-1}</math>. That is to say, the [[commutator]] of <math>r_i</math> and <math>r_j</math> is equal to 1, or equivalently that <math>r_i</math> and <math>r_j</math> commute.

The reason that  <math>m_{ij} = m_{ji}</math> for <math>i \neq j</math> is stipulated in the definition is that
:<math>yy = 1</math>,
together with
:<math>(xy)^m = 1</math>
already implies that
:<math>(yx)^m = (yx)^myy = y(xy)^my = yy = 1</math>. 
An alternative proof of this implication is the observation that <math>(xy)^k</math> and <math>(yx)^k</math> are [[conjugate elements|conjugates]]: indeed <math>y(xy)^k y^{-1} = (yx)^k yy^{-1}=(yx)^k</math>.

===Coxeter matrix and Schläfli matrix===
The '''Coxeter matrix''' is the <math>n\times n</math> [[symmetric matrix]] with entries <math>m_{ij}</math>. Indeed, every symmetric matrix with diagonal entries exclusively 1 and nondiagonal entries in the set <math>\{2,3,\ldots\} \cup \{\infty\}</math> is a Coxeter matrix.

The Coxeter matrix can be conveniently encoded by a '''[[Coxeter–Dynkin diagram|Coxeter diagram]]''', as per the following rules.
* The vertices of the graph are labelled by generator subscripts.
* Vertices <math>i</math> and <math>j</math> are adjacent if and only if <math>m_{ij}\geq 3</math>.
* An edge is labelled with the value of <math>m_{ij}</math> whenever the value is <math>4</math> or greater.

In particular, two generators [[commutative operation|commute]] if and only if they are not joined by an edge. 
Furthermore, if a Coxeter graph has two or more [[connected component (graph theory)|connected component]]s, the associated group is the [[direct product of groups|direct product]] of the groups associated to the individual components.
Thus the [[disjoint union]] of Coxeter graphs yields a [[direct product of groups|direct product]] of Coxeter groups.

The Coxeter matrix, <math>M_{ij}</math>, is related to the <math>n\times n</math> [[Schläfli matrix]] <math>C</math> with entries <math>C_{ij} = -2\cos(\pi/M_{ij})</math>, 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.

{| class=wikitable
|+ Examples
|- align=center
!Coxeter group
! A<sub>1</sub>×A<sub>1</sub>
! A<sub>2</sub>
! B<sub>2</sub>
! I<sub>2</sub>(5)
! G<sub>2</sub>
! <math>{\tilde{A}}_1 = I_2(\infty)</math>
! A<sub>3</sub>
! B<sub>3</sub>
! D<sub>4</sub>
! <math>{\tilde{A}}_3</math>
|- align=center
!Coxeter diagram
|{{CDD|node|2|node}}
|{{CDD|node|3|node}}
|{{CDD|node|4|node}}
|{{CDD|node|5|node}}
|{{CDD|node|6|node}}
|{{CDD|node|infin|node}}
|{{CDD|node|3|node|3|node}}
|{{CDD|node|4|node|3|node}}
|{{CDD|node|3|node|split1|nodes}}
|{{CDD|node|split1|nodes|split2|node}}
|- align=center
!Coxeter matrix
|<math>\left [
\begin{smallmatrix}
 1 & 2 \\
 2 & 1 \\ 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 3 \\
 3 & 1 \\ 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 4 \\
 4 & 1 \\ 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 5 \\
 5 & 1 \\ 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 6 \\
 6 & 1 \\ 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & \infty \\
 \infty & 1 \\ 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 3 & 2 \\
 3 & 1 & 3 \\
 2 & 3 & 1 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 4 & 2 \\
 4 & 1 & 3 \\
 2 & 3 & 1 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 3 & 2 & 2 \\
 3 & 1 & 3 & 3 \\
 2 & 3 & 1 & 2\\
 2 & 3 & 2 & 1 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 1 & 3 & 2 & 3 \\
 3 & 1 & 3 & 2 \\
 2 & 3 & 1 & 3\\
 3 & 2 & 3 & 1 
\end{smallmatrix}\right ]</math>
|- align=center
!Schläfli matrix
|<math>\left [
\begin{smallmatrix}
 2 & 0 \\
 0 & 2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 & -1 \\
 -1 & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 & -\sqrt2 \\
 -\sqrt2 & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 & -\phi \\
 -\phi & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 & -\sqrt3 \\
 -\sqrt3 & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 & -2 \\
 -2 & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 & -1  & \ \,0 \\
 -1  & \ \,2 & -1 \\
 \ \,0 & -1  & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,\ \ 2 & -\sqrt{2} & \ \,0 \\
 -\sqrt{2} & \ \,\ \ 2 &   -1 \\
 \ \,\ \ 0 &   \ \,-1 & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 &  -1 & \ \,0 & \ \,0 \\
  -1 & \ \,2 &  -1 &  -1 \\
 \ \,0 &  -1 & \ \,2 & \ \,0 \\
 \ \,0 &  -1 & \ \,0 & \ \,2 
\end{smallmatrix}\right ]</math>
|<math>\left [
\begin{smallmatrix}
 \ \,2 &  -1 & \ \,0 &  -1 \\
  -1 & \ \,2 &  -1 & \ \,0 \\
 \ \,0 &   -1 & \ \,2 &  -1 \\
  -1 & \ \,0 &  -1 & \ \,2 
\end{smallmatrix}\right ]</math>
|}