Towards a uniform subword complex description of acyclic finite type cluster algebras

TL;DR

This paper advances the understanding of acyclic finite type cluster algebras by describing c- and g-vectors and proposing a conjecture on the Newton polytopes of F-polynomials, linking cluster complexes to Minkowski sums.

Contribution

It introduces a unified subword complex framework for describing key elements of acyclic finite type cluster algebras and proposes a conjecture relating Newton polytopes to cluster complexes.

Findings

Description of c- and g-vectors for acyclic finite type cluster algebras

A conjecture on the Newton polytopes of F-polynomials

Connection between cluster complexes and Minkowski sums of Newton polytopes

Abstract

It has been established in recent years how to approach acyclic cluster algebras of finite type using subword complexes. In this paper, we continue this study by describing the c- and g-vectors, and by providing a conjectured description of the Newton polytopes of the F-polynomials. In particular, we show that this conjectured description would imply that finite type cluster complexes are realized by the duals of the Minkowski sums of the Newton polytopes of either the F-polynomials, or of the cluster variables, respectively.