The Euler characteristic of an endotrivial complex
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908483892281344 |
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| author | Mazza, Nadia Miller, Sam K. |
| author_facet | Mazza, Nadia Miller, Sam K. |
| contents | Let $G$ be a finite group and $k$ a field of prime characteristic $p$. We examine the Lefschetz homomorphism $Λ: \mathcal{E}_k(G) \to O(T(kG))$ from the group of endotrivial complexes, i.e. the Picard group of the bounded homotopy category of $p$-permutation modules $K^b({}_{kG}\mathbf{triv})$, to the orthogonal unit group of the Grothendieck group of $K^b({}_{kG}\mathbf{triv})$, i.e. the trivial source ring. When $p = 2$ and $k = \mathbb{F}_2$, $Λ$ is surjective when $G$ has a Sylow $2$-subgroup with fusion controlled by its normalizer, and when $G$ has dihedral Sylow $2$-subgroups. When $p$ is odd, $Λ$ is surjective if $G$ has a cyclic Sylow $p$-subgroup or is $p$-nilpotent, but we exhibit examples of groups of $p$-rank 2 or greater for which $Λ$ is not surjective. We also examine the kernel of the Lefschetz homomorphism, determining it for all groups when $p = 2$ and for groups with cyclic Sylow $p$-subgroups when $p$ is odd. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07404 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Euler characteristic of an endotrivial complex Mazza, Nadia Miller, Sam K. Representation Theory Group Theory K-Theory and Homology 20J05, 19A22, 20C05, 20C20 Let $G$ be a finite group and $k$ a field of prime characteristic $p$. We examine the Lefschetz homomorphism $Λ: \mathcal{E}_k(G) \to O(T(kG))$ from the group of endotrivial complexes, i.e. the Picard group of the bounded homotopy category of $p$-permutation modules $K^b({}_{kG}\mathbf{triv})$, to the orthogonal unit group of the Grothendieck group of $K^b({}_{kG}\mathbf{triv})$, i.e. the trivial source ring. When $p = 2$ and $k = \mathbb{F}_2$, $Λ$ is surjective when $G$ has a Sylow $2$-subgroup with fusion controlled by its normalizer, and when $G$ has dihedral Sylow $2$-subgroups. When $p$ is odd, $Λ$ is surjective if $G$ has a cyclic Sylow $p$-subgroup or is $p$-nilpotent, but we exhibit examples of groups of $p$-rank 2 or greater for which $Λ$ is not surjective. We also examine the kernel of the Lefschetz homomorphism, determining it for all groups when $p = 2$ and for groups with cyclic Sylow $p$-subgroups when $p$ is odd. |
| title | The Euler characteristic of an endotrivial complex |
| topic | Representation Theory Group Theory K-Theory and Homology 20J05, 19A22, 20C05, 20C20 |
| url | https://arxiv.org/abs/2508.07404 |