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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2112.09257 |
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| _version_ | 1866912904549236736 |
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| author | Nordskova, Anya |
| author_facet | Nordskova, Anya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_09257 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Derived Picard groups of symmetric representation-finite algebras of type $D$ Nordskova, Anya Representation Theory 18G80, 16G20 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. |
| title | Derived Picard groups of symmetric representation-finite algebras of type $D$ |
| topic | Representation Theory 18G80, 16G20 |
| url | https://arxiv.org/abs/2112.09257 |