A recognition theorem for permutation modules over $p$-groups extending Weiss' Theorem
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917102051393536 |
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| author | Estanislau, Marlon |
| author_facet | Estanislau, Marlon |
| contents | Let $G$ be a finite $p$-group with normal subgroup $N$, and $R$ a complete discrete valuation ring in mixed characteristic. We characterize permutation $RG$-modules in terms of modules for $RN$ and $R[G/N]$. The result generalizes both the seminal detection theorem for permutation modules due to Weiss, who characterizes those permutation $RG$-modules that are $RN$-free when $R$ is a finite extension of $\mathbb{Z}_p$, and a more recent result of MacQuarrie and Zalesskii, who prove a characterization of permutation modules when $N$ has order $p$ and $R = \mathbb{Z}_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19710 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A recognition theorem for permutation modules over $p$-groups extending Weiss' Theorem Estanislau, Marlon Representation Theory 20c11 Let $G$ be a finite $p$-group with normal subgroup $N$, and $R$ a complete discrete valuation ring in mixed characteristic. We characterize permutation $RG$-modules in terms of modules for $RN$ and $R[G/N]$. The result generalizes both the seminal detection theorem for permutation modules due to Weiss, who characterizes those permutation $RG$-modules that are $RN$-free when $R$ is a finite extension of $\mathbb{Z}_p$, and a more recent result of MacQuarrie and Zalesskii, who prove a characterization of permutation modules when $N$ has order $p$ and $R = \mathbb{Z}_p$. |
| title | A recognition theorem for permutation modules over $p$-groups extending Weiss' Theorem |
| topic | Representation Theory 20c11 |
| url | https://arxiv.org/abs/2511.19710 |