Newton polytopes in cluster algebras and $\tau$-tilting theory

TL;DR

This paper demonstrates that Newton polytopes of $F$-polynomials uniquely determine cluster monomials and $ au$-rigid modules in skew-symmetrizable cluster algebras and $ au$-tilting theory, revealing a geometric characterization.

Contribution

It establishes a novel link between Newton polytopes and the unique determination of cluster monomials and $ au$-rigid modules, using tools like the left Bongartz completion and $F$-invariants.

Findings

Cluster monomials are uniquely determined by Newton polytopes of $F$-polynomials.

$ au$-rigid modules are uniquely determined by Newton polytopes.

The proofs utilize the left Bongartz completion and $F$-invariants.

Abstract

We prove that the cluster monomials in non-initial cluster variables are uniquely determined by the Newton polytopes of their $F$-polynomials for skew-symmetrizable cluster algebras. Accordingly, we prove that the $\tau$-rigid modules and the left finite multi-semibricks in $\tau$-tilting theory are uniquely determined by the Newton polytopes of these modules. The key tools used in the proofs are the left Bongartz completion, $F$-invariant and partial $F$-invariant in the context of cluster algebras and $\tau$-tilting theory.