Hom-configurations and noncrossing partitions

Abstract

We study maximal Hom-free sets in the τ [2]-orbit category C(Q) of the bounded derived category for the path algebra associated to a Dynkin quiver Q, where τ denotes the Auslander-Reiten translation and [2] denotes the square of the shift functor. 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 mutation of maximal Hom-free sets in C(Q) and prove that the graph of these mutations is connected.

Reductions of triangulated categories and simple‐minded collections

“…Simple‐minded systems (SMSs) in stable module categories were introduced in [30] and studied for negative CY triangulated categories in [15, 17]. Recently, there is increasing interest in negative CY triangulated categories (see, for example, [7, 12–16, 19]), including the stable categories of Cohen–Macaulay (CM) dg modules [21].…”