• root systems
• nestohedra
• group action
• euler characteristic
• nested sets
• building sets

The idea of this thesis moves from the article "Permutonestohedra", in which a family of polytopes, dubbed permutonestohedra is described. These polytopes are related to some important objects in algebra, combinatorics and
geometry: root systems, Weyl groups, building sets and nested sets, compactications of conguration
spaces. We start presenting nestohedra, then we consider their convex hull, that is the permutonestohedron. We come back to nestohedra to consider other group actions and to conclude this thesis we compute the Euler characteristic of the manyfold obtained by identification of nestohedra that are in opposite Weyl chambers.