Proposition 1

Let F be a facet and C a chamber such that $F \subset \overline{C}$. Let $w \in W$. The following conditions are equivalent: (i) $w(F)$ meets F; (ii) $w(F) = F$; (iii) $w(\overline{F}) = \overline{F}$;

(iv) $w$ fixes at least one point of $F$; (v) $w$ fixes every point of $F$; (vi) $w$ fixes every point of $\overline{F}$; (vii) $w$ belongs to the subgroup of $W$ generated by the reflections with respect to the walls of $C$ containing $F$.

For every subset $A$ of $E$, denote by $W(A)$ the subgroup of $W$ consisting of the elements that fix every point of $A$.