Coxeter fan

Let $(W,S)$ be a finite Coxeter system acting by reflections on an $\mathbb{ℝ}$-Euclidean space. Let $\mathit{a}$ be a point in the complement of the hyperplanes corresponding to the reflections in $W$. The convex hull of the $W$-orbit of $\mathit{a}$ is a simple convex polytope: the well-known permutahedron $\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{m}{\mathrm{a}}^{\mathit{a}}(W)$. The normal fan of $\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{m}(W)$ is the Coxeter fan ${ℱ}$.

More generally, given a Weyl group $W$, the Coxeter arrangement ${𝒜}$ for $W$ is the collection of all reflecting hyperplanes for $W$. 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 ${ℱ}$ associated to ${𝒜}$.