Permutonestohedra

TL;DR

This paper introduces permutonestohedra, a new class of polytopes generalizing permutoassociahedra, providing explicit realizations of real spherical models associated with root systems and Coxeter groups, with potential applications in geometric combinatorics.

Contribution

It constructs permutonestohedra as convex hulls of nestohedra within chambers, generalizing known polytopes and linking root systems with geometric models.

Findings

Explicit realization of real spherical models as unions of nestohedra

Definition of permutonestohedra as convex hulls with symmetry group actions

Generalization of Kapranov's permutoassociahedra

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 Phi with finite Coxeter group W and any W-invariant building set, we describe an explicit realization of the real spherical model as a union of polytopes (nestohedra) which 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 W or also, depending on the building set, of Aut(Phi). The permutonestohedra are natural generalizations of Kapranov's permutoassociahedra.