Cambrian lattices are fractionally Calabi-Yau via 2-cluster combinatorics

Abstract

Reading constructed a Cambrian lattice $C_{\Gamma }$ for each oriented finite type Coxeter diagram $\Gamma$. We show that the derived category of representations of $C_{\Gamma }$ is fractionally Calabi-Yau for any $\Gamma$, confirming a conjecture of Chapoton. This extends a result of Rognerud for Cambrian lattices of type $A$ with linear orientation, better known as Tamari lattices. If $\Gamma$ is crystallographic, then $C_{\Gamma }$ is given by the lattice of torsion classes of any hereditary algebra $\Lambda$ of type $\Gamma$. In this case we introduce and study a class of intervals in $C_{\Gamma }$ whose combinatorics matches the combinatorics of $2$-cluster tilting objects in the 2-cluster category of $\Lambda$. This allows us to compute the Calabi-Yau dimension of $C_{\Gamma }$.

With an appendix by Rene Marczinzik