Given an ADE Dynkin diagram, we also have a corresponding Weyl group. For example, the $A_n$ diagrams give the symmetric groups. Applying the Weyl group elements to a generic point in the root space (vector space) you obtain a convex polyhedron. For $A_2$ you obtain a hexagon in the plane.
I'm trying to find a reference where someone has explicitly worked out the combinatorial structure of the corresponding permutahedron, in the case of ADE root systems.
For example, what are the various facets and their containment relations? What is the volume? How many lattice points does it contain? These are questions stated in the following lecture (https://youtu.be/FgOGTGhhe48?list=PLBF39AFBBC3FB30AF&t=1760).