Definition 1. A Coxeter group is a group with presentation
where and for Also says that for any
To be clear, the above is technically finitely presented. Although we could have Coxeter groups with an infinite number of generators.
Proposition 2. Let be a Coxeter group with generators Then we have
- If for all distinct then is abelian;
Proof. (i) Trivial since (ii) We have We also of course have that Thus
Thus The other inequality is dually shown.
Let be the matrix defined by This is called the Coxeter matrix of
Let be a Coxeter group with generators and define an inner product on
Let such that Then define
We then define the representation of the Coxeter group on by
We also assume goes to Let us now show that the relations carry over: First suppose then
For note that is acting on the subspace One can show that has order on this subspace.
 Abramenko, Peter and Kenneth Brown. Buildings. Graduate Texts in Mathematics. Vol. 248. Springer Science and Business Media. 2008.