Bongartz completion via $c$-vectors

TL;DR

This paper characterizes Bongartz completion via $c$-vectors in $ au$-tilting theory and extends the concept to cluster algebras, proving existence, uniqueness, and applications related to exchange quivers and cluster Poisson seeds.

Contribution

It introduces a new characterization of Bongartz completion using $c$-vectors and establishes its existence, uniqueness, and properties in cluster algebras.

Findings

Bongartz completion in cluster algebras is well-defined and unique.

Certain subquivers of exchange quivers are isomorphic to other exchange quivers.

Cluster Poisson seeds are uniquely determined by negative variables.

Abstract

In the present paper, we first give a characterization for Bongartz completion in $\tau$-tilting theory via $c$-vectors. Motivated by this characterization, we give the definition of Bongartz completion in cluster algebras using $c$-vectors. Then we prove the existence and uniqueness of Bongartz completion in cluster algebras. We also prove that Bongartz completion admits certain commutativity. We give two applications for Bongartz completion in cluster algebras. As the first application, we prove the full subquiver of the exchange quiver (or known as oriented exchange graph) of a cluster algebra $\mathcal A$ whose vertices consist of seeds of $\mathcal A$ containing particular cluster variables is isomorphic to the exchange quiver of another cluster algebra. As the second application, we prove that in a cluster Poisson algebra $\mathcal X_\bullet$, each cluster Poisson seed (up to seed equivalence) of $\mathcal X_\bullet$ is uniquely determined by its negative cluster Poisson variables.