Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type A_n

Abstract

We give a complete classification of torsion pairs in the cluster category of Dynkin type A_n. Along the way we give a new combinatorial description of Ptolemy diagrams, an infinite version of which was introduced by Ng. This allows us to count the number of torsion pairs in the cluster category of type A_n. We also count torsion pairs up to Auslander-Reiten translation.

14 pages; v2: Major simplification of the formula in Theorem B for the number of Ptolemy diagrams; v3: 17 pages; revision following referee's very helpful comments; more details for some arguments added; formula in Prop. 3.5 (formerly 3.4) corrected; final version, to appear in J. Algebraic Combin