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