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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.15348 |
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| _version_ | 1866908599266050048 |
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| author | Li, Liping |
| author_facet | Li, Liping |
| contents | Given an infinite set $Ω$ and a ring $R$ as well as a group $G$ acting on them, we show that $G$ and a subgroup $H$ share the same canonical relational structure on $Ω$ if and only if the restriction functor gives an equivalence from the category of discrete representations of $G$ to that of $H$. Moreover, the age of this relational structure satisfies the strong amalgamation property if and only if there is a canonical isomorphism from the category of finite substructures of $Ω$ and embeddings to the opposite category of the orbit category of $G$. As an application, we prove that finitely generated discrete representations of highly homogeneous groups over the polynomial ring $k[Ω]$ are Noetherian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15348 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On representations of permutation groups and orbit categories Li, Liping Representation Theory Given an infinite set $Ω$ and a ring $R$ as well as a group $G$ acting on them, we show that $G$ and a subgroup $H$ share the same canonical relational structure on $Ω$ if and only if the restriction functor gives an equivalence from the category of discrete representations of $G$ to that of $H$. Moreover, the age of this relational structure satisfies the strong amalgamation property if and only if there is a canonical isomorphism from the category of finite substructures of $Ω$ and embeddings to the opposite category of the orbit category of $G$. As an application, we prove that finitely generated discrete representations of highly homogeneous groups over the polynomial ring $k[Ω]$ are Noetherian. |
| title | On representations of permutation groups and orbit categories |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2510.15348 |