Permutonestohedra

Abstract

There are several real spherical models associated with a root arrangement, depending on the choice of a building set. The connected components of these models are manifolds with corners which can be glued together to obtain the corresponding real De Concini–Procesi models. In this paper, starting from any root system (Formula presented.) with finite Coxeter group (Formula presented.) and any (Formula presented.)-invariant building set, we describe an explicit realization of the real spherical model as a union of polytopes (nestohedra) that lie inside the chambers of the arrangement. The main point of this realization is that the convex hull of these nestohedra is a larger polytope, a permutonestohedron, equipped with an action of (Formula presented.) or also, depending on the building set, of (Formula presented.). The permutonestohedra are natural generalizations of Kapranov’s permutoassociahedra.