Torsion pairs in repetitive cluster categories of type An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_n$$\end{document}

We give a complete classification of torsion pairs in repetitive cluster categories of type An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_n$$\end{document}, which were defined by Zhu (Comm Algebra 39:2437–2448, 2011) as the orbit categories Db(modKAn)/⟨(τ-1[1])p⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^b(\mathop {\textrm{mod}}\nolimits KA_n)/\langle (\tau ^{-1}[1])^p\rangle $$\end{document}, via certain configurations of diagonals, called Ptolemy diagrams. As applications, we classify rigid subcategories of Db(modKAn)/⟨(τ-1[1])p⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^b(\mathop {\textrm{mod}}\nolimits KA_n)/\langle (\tau ^{-1}[1])^p\rangle $$\end{document}, which gives Lamberti’s classification of cluster tilting subcategories (Lamberti in J Algebra Appl 13:1350091, 2014). When p=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=1$$\end{document}, this generalizes the work of Holm, Jørgensen and Rubey for the classification of torsion pairs in cluster categories of type An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{n}$$\end{document} (Holm et al., J. Algebraic Combin 34:507–523, 2011).