ABHY Associahedra and Newton polytopes of $F$-polynomials for finite type cluster algebras

TL;DR

This paper demonstrates that a recent associahedron construction can be generalized to all simply-laced Dynkin types and also produces Newton polytopes for F-polynomials in cluster algebras, linking geometry and algebra.

Contribution

It extends a new associahedron construction to all simply-laced types and reveals its connection to Newton polytopes of F-polynomials in cluster algebras.

Findings

Construction applies to all simply-laced Dynkin types

Produces Newton polytopes for F-polynomials

Toric variety's nef cone is simplicial

Abstract

A new construction of the associahedron was recently given by Arkani-Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct generalized associahedra of any simply-laced Dynkin type. Unexpectedly, we also show that this same construction produces Newton polytopes for all the $F$-polynomials of the corresponding cluster algebras. In addition, we show that the toric variety associated to the g-vector fan has the property that its nef cone is simplicial.