Brauer pairs for splendid Rickard equivalences
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866911302644924416 |
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| author | Breland, Jadyn V. Miller, Sam K. |
| author_facet | Breland, Jadyn V. Miller, Sam K. |
| contents | We define the notion of a Brauer pair of a chain complex, extending the notion of a Brauer pair of a $p$-permutation module introduced by Boltje and Perepelitsky. In fact, the Brauer pairs of a splendid Rickard equivalence $C$ coincide with the set of Brauer pairs of the corresponding $p$-permutation equivalence $Λ(C)$ induced by $C$. As a result, we derive structural results for splendid Rickard equivalences that correspond to known structural properties for $p$-permutation equivalences. In particular, we show splendid Rickard equivalences induce local splendid Rickard equivalences between normalizer block algebras as well as centralizer block algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10258 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Brauer pairs for splendid Rickard equivalences Breland, Jadyn V. Miller, Sam K. Representation Theory Group Theory 20C05, 20C20, 19A22, 20J05 We define the notion of a Brauer pair of a chain complex, extending the notion of a Brauer pair of a $p$-permutation module introduced by Boltje and Perepelitsky. In fact, the Brauer pairs of a splendid Rickard equivalence $C$ coincide with the set of Brauer pairs of the corresponding $p$-permutation equivalence $Λ(C)$ induced by $C$. As a result, we derive structural results for splendid Rickard equivalences that correspond to known structural properties for $p$-permutation equivalences. In particular, we show splendid Rickard equivalences induce local splendid Rickard equivalences between normalizer block algebras as well as centralizer block algebras. |
| title | Brauer pairs for splendid Rickard equivalences |
| topic | Representation Theory Group Theory 20C05, 20C20, 19A22, 20J05 |
| url | https://arxiv.org/abs/2312.10258 |