Polytopal realizations of finite type $\mathbf{g}$-vector fans

TL;DR

This paper proves that all finite type g-vector fans can be realized as normal fans of polytopes, constructing a universal associahedron for each Dynkin type that encompasses all such fans.

Contribution

It introduces a universal associahedron for each finite Dynkin type, demonstrating the polytopality of all finite type g-vector fans, regardless of acyclicity.

Findings

Finite type g-vector fans are polytopal.

Construction of a universal associahedron for each Dynkin type.

Any g-vector fan of a given type is a projection of this universal polytope.

Abstract

This paper shows the polytopality of any finite type $\mathbf{g}$-vector fan, acyclic or not. In fact, for any finite Dynkin type $\Gamma$, we construct a universal associahedron $\mathsf{Asso}_{\mathrm{un}}(\Gamma)$ with the property that any $\mathbf{g}$-vector fan of type $\Gamma$ is the normal fan of a suitable projection of $\mathsf{Asso}_{\mathrm{un}}(\Gamma)$.