Cluster algebras of type C via cluster tubes

Details Cluster algebras of type C via cluster tubes Cluster algebras of type C via cluster tubes

2010-08-20

arxiv.org/abs/1008.3444v1

Mathematics

Representation Theory

Scientific paper

Cluster tubes are $2-$Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a maximal rigid object $T$ in the cluster tube $\C_n$ of rank $n$. For any indecomposable rigid object $M$ in $\C_n$, we give an analogous $X_M$ of Caldero-Chapton's formula (or Palu's cluster character formula) by using the geometric information of $M$. We show that $X_M, X_{M'}$ satisfy the mutation formula when $M,M'$ form an exchange pair, and that $X_{?}: M\mapsto X_M$ gives a bijection from the set of indecomposable rigid objects in $\C_n$ to the set of cluster variables of cluster algebra of type $C_{n-1}$, which induces a bijection between the set of (basic) maximal rigid objects in $\C_n$ and the set of clusters. This strengths a surprising result proved recently by Buan-Marsh-Vatne that the combinatorics of maximal rigid objects in the cluster tube $\C_n$ encode the combinatorics of the cluster algebra of type $B_{n-1}$ since the combinatorics of cluster algebras of type $B_{n-1}$ or of type $C_{n-1}$ are the same by one of results of Fomin-Zelevinsky in [FZ2].

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