**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

(iii) so

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.

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