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

TL;DR

This paper proves that the bounded derived category of the incidence algebra of Tamari lattices is fractionally Calabi-Yau, confirming a conjecture and using combinatorial methods to describe the Serre functor.

Contribution

It establishes the fractional Calabi-Yau property for Tamari lattice derived categories and provides a combinatorial description of the Serre functor.

Findings

The derived category of Tamari lattices is fractionally Calabi-Yau.

A combinatorial description of the Serre functor is provided.

Confirms a conjecture of Chapoton.

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.