Tagged mapping class groups: Auslander-Reiten translation

Abstract

We give a geometric realization, the tagged rotation, of the AR-translation on the generalized cluster category associated to a surface $\mathbf{S}$ with marked points and non-empty boundary, which generalizes Brüstle-Zhang's result for the puncture free case. As an application, we show that the intersection of the shifts in the 3-Calabi-Yau derived category $\mathcal{D}({\Gamma }_{\mathbf{S}})$ associated to the surface and the corresponding Seidel-Thomas braid group of $\mathcal{D}({\Gamma }_{\mathbf{S}})$ is empty, unless $\mathbf{S}$ is a polygon with at most one puncture (i.e. of type A or D).

To appear in Mathematische Zeitschrift

Cited by in corpus (7)

• Cluster categories for marked surfaces: punctured case
• Decorated marked surfaces: spherical twists versus braid twists
• Density of $g$-vector cones from triangulated surfaces
• Cluster automorphism groups of cluster algebras of finite type
• Cluster automorphism groups of cluster algebras with coefficients
• Minimal length maximal green sequences
• Denominator vectors and dimension vectors from triangulated surfaces