Hom-configurations and noncrossing partitions.

Coelho Simões, Raquel

Journal of Algebraic Combinatorics, Tome 35 (2012) no. 2, pp. 313-343.

Résumé

Summary: We study maximal Hom-free sets in the $\tau $[2]-orbit category $C( Q)$ of the bounded derived category for the path algebra associated to a Dynkin quiver $Q$, where $\tau $ 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 $_{\leq 0}$-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) \mathcal $C$(Q)$ and prove that the graph of these mutations is connected.

Détail

Keywords: keywords cluster combinatorics, derived category, exceptional sequences, hom-configurations, mutations, noncrossing partitions, perpendicular categories, reflection functors, quiver representations