On the bounded derived category of the Tamari lattices.
Tamari lattices are nice classical combinatorial partial orders that appear in several algebraic contexts. In this talk I will consider the bounded derived category of the incidence algebras of these posets and present various results and conjectures relating deep combinatorial properties of these posets and the algebraic structure of this derived category.
We will focus on a conjecture of Chapoton involving Serre functors of a bounded derived category of the incidence algebra of the Tamari lattice. We will show that there is a nice combinatorial description, in terms of noncrossing trees, of the action of these Serre functors on a well chosen family of objects.