Cluster algebras arising from cluster tubes I: integer vectors

TL;DR

This paper explores the structure of cluster algebras from cluster tubes, providing categorical interpretations for key vectors and proving a denominator theorem that confirms linear independence of certain cluster variables.

Contribution

It introduces categorical interpretations for $g$-vectors, $c$-vectors, and denominator vectors in cluster algebras of type C, and proves a new denominator theorem.

Findings

Categorical interpretations for $g$-vectors, $c$-vectors, and denominator vectors.

Proof of a denominator theorem ensuring linear independence of denominator vectors.

Strengthens the connection between cluster tubes and type C cluster algebras.

Abstract

We study cluster algebras arising from cluster tubes. We obtain categorical interpretations for $g$-vectors, $c$-vectors and denominator vectors for cluster algebras of type $\mathrm{C}$ with respect to arbitrary initial seeds. In particular, a denominator theorem has been proved, which enables us to establish the linearly independence of denominator vectors of cluster variables from the same cluster for cluster algebras of type $\mathrm{A}\mathrm{B}\mathrm{C}$. This strengthens the link between cluster tubes and cluster algebras of type $\mathrm{C}$ initiated by Buan, Marsh and Vatne.