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

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.

Figures (8)

• Figure 1: Obtaining $Q_{G^\gamma}$ from $Q_G$.
• Figure 2: Constructing the modified Brauer tree $G^\gamma$ of $A_{G^\gamma}$ (on the right) from $G$ (on the left).
• Figure 3: Construction of the quiver $Q'$ from the three quivers $Q^1, Q^2$ and $Q^3$.
• Figure 4: The modified Brauer tree $G'$ of $A_{Q'}$ (on the left) and an alternative way to depict it using 2-cells (on the right).
• Figure 17: The quiver $Q_m$ (on the left) and the modified Brauer tree $\Gamma_m$ (on the right) of $\Lambda_m$.

Theorems & Definitions (40)

• Definition 2.1
• Remark
• Theorem 1: Rickard, R, R2
• Remark
• Definition 2.2: Rouquier, Zimmermann RZ, Yekutieli Y
• Definition 2.3
• Definition 2.4
• Remark
• Theorem 2: Aihara, Aih
• Definition 2.5