Derived Picard groups of symmetric representation-finite algebras of type $D$

TL;DR

This paper explicitly describes the derived Picard groups of symmetric representation-finite algebras of type D, showing they are generated by spherical twists, shifts, and automorphisms, using combinatorial models and braid group actions.

Contribution

It provides a detailed description of the derived Picard groups for type D symmetric algebras, introducing a combinatorial-geometric model for silting mutations and proving the faithfulness of certain braid group actions.

Findings

Derived Picard groups are generated by spherical twists, shifts, and automorphisms.

Developed a combinatorial-geometric model for silting mutations in type D.

Proved the faithfulness of the braid group action via spherical twists.

Abstract

We explicitly describe the derived Picard groups of symmetric representation-finite algebras of type $D$. In particular, we prove that these groups are generated by spherical twists along collections of $0$-spherical objects, the shift and autoequivalences which come from outer automorphisms of a particular representative of the derived equivalence class. The arguments we use are based on the fact that symmetric representation-finite algebras are tilting-connected. To apply this result we in particular develop a combinatorial-geometric model for silting mutations in type $D$, generalising the classical concepts of Brauer trees and Kauer moves. Another key ingredient in the proof is the faithfulness of the braid group action via spherical twists along $D$-configurations of $0$-spherical objects.