Positivity for cluster algebras of rank 3

TL;DR

This paper proves the positivity conjecture for a specific class of rank 3 cluster algebras, confirming that all cluster variables can be expressed with positive coefficients.

Contribution

It establishes the positivity conjecture for skew-symmetric coefficient-free cluster algebras of rank 3, a previously unresolved case.

Findings

Positivity conjecture proven for rank 3 cases

All cluster variables have positive Laurent expansions

Advances understanding of cluster algebra structure

Abstract

We prove the positivity conjecture for skew-symmetric coefficient-free cluster algebras of rank 3.