Coxeter fan

Let [math]\displaystyle{ (W, S) }[/math] be a finite Coxeter system acting by reflections on an [math]\displaystyle{ \mathbb{R} }[/math]-Euclidean space. Let [math]\displaystyle{ \boldsymbol{a} }[/math] be a point in the complement of the hyperplanes corresponding to the reflections in [math]\displaystyle{ W }[/math]. The convex hull of the [math]\displaystyle{ W }[/math]-orbit of [math]\displaystyle{ \boldsymbol{a} }[/math] is a simple convex polytope: the well-known permutahedron [math]\displaystyle{ \mathrm{Perm}\operatorname{a}^{\boldsymbol{a}}(W) }[/math]. The normal fan of [math]\displaystyle{ \mathrm{Perm}(W) }[/math] is the Coxeter fan [math]\displaystyle{ \mathcal{F} }[/math].

More generally, given a Weyl group [math]\displaystyle{ W }[/math], the Coxeter arrangement [math]\displaystyle{ \mathcal{A} }[/math] for [math]\displaystyle{ W }[/math] is the collection of all reflecting hyperplanes for [math]\displaystyle{ W }[/math]. The complement of the arrangement consists of open cones, whose closures are called chambers. The collection of chambers and all of their faces define the Coxeter fan [math]\displaystyle{ \mathcal{F} }[/math] associated to [math]\displaystyle{ \mathcal{A} }[/math].