Cluster algebras arising from cluster tubes II: the Caldero-Chapoton map

Abstract

We continue our investigation on cluster algebras arising from cluster tubes. Let $\mathcal{C}$ be a cluster tube of rank $n+1$. For an arbitrary basic maximal rigid object $T$ of $\mathcal{C}$, one may associate a skew-symmetrizable integer matrix $B_T$ and hence a cluster algebra $\mathcal{A}(B_T)$ to $T$. We define an analogue Caldero-Chapoton map ${\mathbb{X}}_M^T$ for each indecomposable rigid object $M\in \mathcal{C}$ and prove that ${\mathbb{X}}_{?}^T$ yields a bijection between the indecomposable rigid objects of $\mathcal{C}$ and the cluster variables of the cluster algebra $\mathcal{A}(B_T)$. The construction of the Caldero-Chapoton map involves Grassmanians of locally free submodules over the endomorphism algebra of $T$. We also show that there is a non-trivial ${\mathbb{C}}^{\times }$-action on the Grassmanians of locally free submodules, which is of independent interest.

26 pages. This is the second part of the lengthy paper arXiv:1801.00709 which was split into two papers. The argument is simplified which is more accessible. version 2: minor changes. Clarified references for the triangle structure of cluster tubes