The enough $g$-pairs property and denominator vectors of cluster algebras

TL;DR

This paper introduces the enough g-pairs property for principal coefficients cluster algebras, proves its universality, and applies it to confirm positivity of denominator vectors and solve longstanding conjectures in the field.

Contribution

It establishes the enough g-pairs property for all skew-symmetrizable principal coefficient cluster algebras and applies it to prove positivity and conjectures about denominator vectors.

Findings

Proves the enough g-pairs property for all skew-symmetrizable principal coefficient cluster algebras.

Confirms positivity of denominator vectors for all skew-symmetrizable cluster algebras.

Provides criteria for identifying whether specific cluster variables belong to the same cluster.

Abstract

In this paper, we introduce the enough $g$-pairs property for a principal coefficients cluster algebra, which can be understood as a strong version of the sign-coherence of the $G$-matrices. Then we prove that any skew-symmetrizable principal coefficients cluster algebra has the enough $g$-pairs property. As an application, we prove the positivity of denominator vectors for any skew-symmetrizable cluster algebra. In fact, we give complete answers to some long standing conjectures on denominator vectors of cluster variables (see Conjecture 1.1 below), which are proposed by Fomin and Zelevinsky in [Compos. Math. 143(2007), 112-164]. In addition, we prove that the seeds whose clusters contain particular cluster variables form a connected subgraph of the exchange graph of this cluster algebra. Lastly, a criterion to distinguish whether particular cluster variables belong to one common cluster is given.