Geometric representation of a Coxeter group
= Geometric representation of a Coxeter group
{c}
{wiki=Coxeter_group#Geometric_representation}
For a finite-rank <Coxeter system>, let $V$ have basis $(e_i)_{i\in I}$ and symmetric <bilinear form>
$$
\langle e_i,e_j\rangle=-2\cos(\pi/m_{ij}).
$$
Its geometric representation sends the generator $x_i$ to the reflection
$$
\sigma(x_i)v=v-\langle v,e_i\rangle e_i.
$$