# Canonical Representation of Finitely Presented Coxeter Groups

Definition 1.  A Coxeter group is a group with presentation

$\langle r_1,...,r_n:(r_ir_j)^{m_{ij}}=1\rangle$

where $m_{ii}=1$ and $m_{ij}\geq 2$ for $i\neq j.$  Also $m_{ij}=\infty$ says that $(r_ir_j)^n\neq 1$ for any $n.$

To be clear, the above is technically finitely presented.  Although we could have Coxeter groups with an infinite number of generators.

Proposition 2.  Let $G$ be a Coxeter group with generators $r_1,...,r_n.$  Then we have

1. $r_i=r_i^{-1}$;
2. If $m_{ij}=2$ for all distinct $i,j,$ then $G$ is abelian;
3. $m_{ij}=m_{ji}$.

Proof.  (i)  Trivial since $m_{ii}=1.$  (ii)  We have $(r_ir_j)^2=1.$  We also of course have that $r_i^2=r_j^2=1.$  Thus

$r_ir_j=r_i(r_ir_j)(r_ir_j)r_j=r_i^2r_jr_ir_j^2=r_jr_i.$

(iii)  $(r_ir_j)^{m_{ij}}=1$ so

$(r_jr_i)^{m_{ij}}=(r_jr_i)^{m_{ij}}r_j^2=r_j(r_ir_j)^{m_{ij}}r_j=1.$

Thus $m_{ji}\leq m_{ij}.$  The other inequality is dually shown.

Let $M$ be the matrix defined by $M_{ij}=m_{ij}.$  This is called the Coxeter matrix of $G.$

Let $G$ be a Coxeter group with $n$ generators and define an inner product on $\mathbb{R}^n$

$\displaystyle\langle e_i,e_j\rangle_G=-\cos\left(\frac{\pi}{m_{ij}}\right).$

Let $a\in\mathbb{R}^n$  such that $\langle a,a\rangle_G\neq 0.$  Then define

$s_a(x)=x-2\langle a,x\rangle_Ga.$

We then define the representation $\rho:G\to L(\mathbb{R}^n)$ of the Coxeter group on $\mathbb{R}^n$ by

$\rho(r_i)=s_{e_i}.$

We also assume $1$ goes to $1.$  Let us now show that the relations carry over:  $(s_{e_i}s_{e_j})^{m_{ij}}=1.$  First suppose $i=j,$ then

$\begin{array}{lcl}(s_{e_i}s_{e_i})^{m_{ii}}(x)&=&s_{e_i}s_{e_i}(x)\\&=&s_{e_i}(x-2\langle e_i,x\rangle_Ge_i)\\&=&x-2\langle e_i,x\rangle_Ge_i-2\langle e_i,x-2\langle e_i,x\rangle_Ge_i\rangle_G e_i\\&=&x-4\langle e_i,x\rangle_Ge_i+4\langle e_i,x\rangle_G\langle e_i,e_i\rangle_Ge_i\\&=&x.\end{array}$

For $i\neq j,$ note that $s_{e_i}s_{e_j}$ is acting on the subspace $\mathbb{R}e_i\oplus\mathbb{R}e_j.$  One can show that $s_{e_i}s_{e_j}$ has order $m_{ij}$ on this subspace.

[1]  Abramenko, Peter and Kenneth Brown.  Buildings.  Graduate Texts in Mathematics.  Vol. 248.  Springer Science and Business Media.  2008.