Endotrivial complexes
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914142860869632 |
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| author | Miller, Sam K. |
| author_facet | Miller, Sam K. |
| contents | Let $G$ be a finite group, $p$ a prime, and $k$ a field of characteristic $p$. We introduce the notion of an endotrivial chain complex of $p$-permutation $kG$-modules, which are the invertible objects in the bounded homotopy category of $p$-permutation $kG$-modules, and study the corresponding Picard group $\mathcal{E}_k(G)$ of endotrivial complexes. Such complexes are shown to induce splendid Rickard autoequivalences of $kG$. The elements of $\mathcal{E}_k(G)$ are determined uniquely by integral invariants arising from the Brauer construction and a degree one character $G \to k^\times$. Using ideas from Bouc's theory of biset functors, we provide a canonical decomposition of $\mathcal{E}_k(G)$, and as an application, give complete descriptions of $\mathcal{E}_k(G)$ for abelian groups and $p$-groups of normal $p$-rank 1. Taking Lefschetz invariants of endotrivial complexes induces a group homomorphism $Λ: \mathcal{E}_k(G) \to O(T(kG))$, where $O(T(kG))$ is the orthogonal unit group of the trivial source ring. Using recent results of Boltje and Carman, we give a Frobenius stability condition elements in the image of $Λ$ must satisfy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_12138 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Endotrivial complexes Miller, Sam K. Representation Theory Group Theory 20J05, 19A22, 20C05, 20C20 Let $G$ be a finite group, $p$ a prime, and $k$ a field of characteristic $p$. We introduce the notion of an endotrivial chain complex of $p$-permutation $kG$-modules, which are the invertible objects in the bounded homotopy category of $p$-permutation $kG$-modules, and study the corresponding Picard group $\mathcal{E}_k(G)$ of endotrivial complexes. Such complexes are shown to induce splendid Rickard autoequivalences of $kG$. The elements of $\mathcal{E}_k(G)$ are determined uniquely by integral invariants arising from the Brauer construction and a degree one character $G \to k^\times$. Using ideas from Bouc's theory of biset functors, we provide a canonical decomposition of $\mathcal{E}_k(G)$, and as an application, give complete descriptions of $\mathcal{E}_k(G)$ for abelian groups and $p$-groups of normal $p$-rank 1. Taking Lefschetz invariants of endotrivial complexes induces a group homomorphism $Λ: \mathcal{E}_k(G) \to O(T(kG))$, where $O(T(kG))$ is the orthogonal unit group of the trivial source ring. Using recent results of Boltje and Carman, we give a Frobenius stability condition elements in the image of $Λ$ must satisfy. |
| title | Endotrivial complexes |
| topic | Representation Theory Group Theory 20J05, 19A22, 20C05, 20C20 |
| url | https://arxiv.org/abs/2309.12138 |