Generic bases for cluster algebras from the cluster category

TL;DR

This paper constructs a candidate basis for (upper) cluster algebras with coefficients from quivers using Hom-finite cluster categories, proving linear independence under certain conditions and connecting to existing bases.

Contribution

It introduces a new basis for cluster algebras derived from cluster categories and establishes its properties and relation to existing conjectures.

Findings

The set of generic values forms a basis when the matrix is full rank.

The basis coincides with known bases in specific cases.

The approach aligns with Fock-Goncharov's conjectures on parametrization.

Abstract

Inspired by recent work of Geiss-Leclerc-Schroer, we use Hom-finite cluster categories to give a good candidate set for a basis of (upper) cluster algebras with coefficients arising from quivers. This set consists of generic values taken by the cluster character on objects having the same index. If the matrix associated to the quiver is of full rank, then we prove that the elements in this set are linearly independent. If the cluster algebra arises from the setting of Geiss-Leclerc-Schroer, then we obtain the basis found by these authors. We show how our point of view agrees with the spirit of conjectures of Fock-Goncharov concerning the parametrization of a basis of the upper cluster algebra by points in the tropical X-variety.