Reflection group of a root system ($W(\Phi )$)

The reflection group of a root system $\Phi$ is the subgroup of the orthogonal group generated by the reflections $s_{\alpha }$ for $\alpha \in \Phi$. It permutes the roots and acts freely and transitively on the fundamental systems.

The Weyl group is generated by the reflections in the roots.

The Coxeter length $\ell (w)$ is the smallest number of simple reflections whose product is $w$. Its parity gives the sign of a Weyl-group element.

If $\lambda$ and $w\lambda$ both lie in the closed dominant chamber, then $w$ is a product of simple reflections whose walls contain $\lambda$. In particular, $w\lambda =\lambda$.

For a chosen positive system of a root system $\Pi$, the inversion set isFor a finite Weyl group, $|N(w)|=\ell (w)$.

In the geometric representation of a Coxeter system,andReplacing “positive” by “negative” reverses either length inequality.