Root Polytopes of Crystallographic Root Systems

Let Phi be a finite (reduced) irreducible crystallographic root system. We give a case-free explicit description of the convex hull of all roots in Phi, that we denote by P-Phi and call the root polytope of Phi. This description is attained by considering a set of distinguished faces, indexed by the subsets of a fixed root basis Pi, which is a complete set of representatives of the orbits of the faces under the action of the Weyl group W. The description reveals a rich combinatorial structure of the root polytope P-Phi and gives as by-products some results on root systems which may be interesting on their own. Even if the proofs (which also are case-free) are clearly omitted, the results are presented in the order they are proved. This is a report on Cellini (Int. Math. Res. Not. 12, 4392-4420 (2015); J. Algebr. Comb. 39(3), 607645 (2014)).