The bounded derived categories of the Tamari lattices are fractionally Calabi-Yau

Abstract

We prove that the bounded derived category of the incidence algebra of the Tamari lattice is fractionally Calabi-Yau, giving a positive answer to a conjecture of Chapoton. The proof involves a combinatorial description of the Serre functor of this derived category on a sufficiently nice family of indecomposable objects.

Comments are welcome. V2: Fixing some types. I made several simplifications for the section on the duality of noncrossing trees