The classification of endotrivial complexes
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911254428254208 |
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| author | Miller, Sam K. |
| author_facet | Miller, Sam K. |
| contents | Let $G$ be a finite group and $k$ a field of prime characteristic $p$. We give a complete classification of endotrivial complexes, i.e. determine the Picard group $\mathcal{E}_k(G)$ of the tensor-triangulated category $K^b({}_{kG}\mathbf{triv})$, the bounded homotopy category of $p$-permutation modules, which Balmer and Gallauer recently considered. For $p$-groups, we identify $\mathcal{E}_k(-)$ with the rational $p$-biset functor $CF_b(-)$ of Borel-Smith functions and recover a short exact sequence of rational $p$-biset functors constructed by Bouc and Yalçin. As a consequence, we prove that every $p$-permutation autoequivalence of a $p$-group arises from a splendid Rickard autoequivalence. Additionally, we give a positive answer to a question of Gelvin and Yalçin, showing the kernel of the Bouc homomorphism for an arbitrary finite group $G$ is described by superclass functions $f: s_p(G) \to \mathbb{Z}$ satisfying the oriented Artin-Borel-Smith conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04088 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The classification of endotrivial complexes Miller, Sam K. Group Theory Representation Theory 20J05, 19A22, 20C05, 20C20 Let $G$ be a finite group and $k$ a field of prime characteristic $p$. We give a complete classification of endotrivial complexes, i.e. determine the Picard group $\mathcal{E}_k(G)$ of the tensor-triangulated category $K^b({}_{kG}\mathbf{triv})$, the bounded homotopy category of $p$-permutation modules, which Balmer and Gallauer recently considered. For $p$-groups, we identify $\mathcal{E}_k(-)$ with the rational $p$-biset functor $CF_b(-)$ of Borel-Smith functions and recover a short exact sequence of rational $p$-biset functors constructed by Bouc and Yalçin. As a consequence, we prove that every $p$-permutation autoequivalence of a $p$-group arises from a splendid Rickard autoequivalence. Additionally, we give a positive answer to a question of Gelvin and Yalçin, showing the kernel of the Bouc homomorphism for an arbitrary finite group $G$ is described by superclass functions $f: s_p(G) \to \mathbb{Z}$ satisfying the oriented Artin-Borel-Smith conditions. |
| title | The classification of endotrivial complexes |
| topic | Group Theory Representation Theory 20J05, 19A22, 20C05, 20C20 |
| url | https://arxiv.org/abs/2403.04088 |