Torsion pairs in repetitive cluster categories of type \(A_n\)

We give a complete classification of torsion pairs in repetitive cluster categories of type \(A_n\) $A_n$ , which were defined by Zhu (Comm Algebra 39:2437–2448, 2011) as the orbit categories \(D^b(\mathop {\textrm{mod}}\nolimits KA_n)/\langle (\tau ^{-1}[1])^p\rangle \) $D^b(\text{mod}K{A}_n)/⟨{({\tau }^{-1}[1])}^p⟩$ , via certain configurations of diagonals, called Ptolemy diagrams. As applications, we classify rigid subcategories of \(D^b(\mathop {\textrm{mod}}\nolimits KA_n)/\langle (\tau ^{-1}[1])^p\rangle \) $D^b(\text{mod}K{A}_n)/⟨{({\tau }^{-1}[1])}^p⟩$ , which gives Lamberti’s classification of cluster tilting subcategories (Lamberti in J Algebra Appl 13:1350091, 2014). When \(p=1\) $p=1$ , this generalizes the work of Holm, Jørgensen and Rubey for the classification of torsion pairs in cluster categories of type \(A_{n}\) $A_n$ (Holm et al., J. Algebraic Combin 34:507–523, 2011).