Hom-configurations and noncrossing partitions

Abstract

Let Q be a Dynkin quiver. The bounded derived category of the path algebra of Q has an autoequivalence given by the composition of the Auslander-Reiten translate and the square of the shift functor. We study maximal Hom-free sets in the corresponding orbit category C(Q). We prove that these sets are in bijection with periodic combinatorial configurations, as introduced by Riedtmann, certain Hom<=0-configurations, studied by Buan, Reiten and Thomas, and noncrossing partitions of the Coxeter group associated to Q which are not contained in any proper standard parabolic subgroup. Note that Reading has proved that these noncrossing partitions are in bijection with positive clusters in the associated cluster algebra. Finally, we give a definition of mutations of maximal Hom-free sets in C(Q) and prove that the graph of these mutations is connected.