Weyl group

The Weyl group $W_R$ of a root system $R\subset E$, where $E$ is a Euclidean vector space,is the subgroup of $\mathrm{GL}(E)$ generated by reflection in the hyperplanes perpendicularto the roots. The map of reflection in a root $\alpha$ is given by

$$r_{\alpha }(v)=v-2\frac{(\alpha ,v)}{(\alpha ,\alpha )}\alpha .$$

The Weyl group is generated by reflections in the simple roots for any choice of a set of positive roots.There is a well-defined length function $\mathrm{\ell }:W_R\to \mathbb{Z}$, where $\mathrm{\ell }(w)$ is the minimalnumber of reflections in simple roots that $w$ can be written as. This is also the number of positiveroots that $w$ takes to negative roots.