On representations of permutation groups and orbit categories
Liping Li
TL;DR
This work establishes when two permutation groups acting on the same infinite set yield equivalent categories of discrete representations via restriction, tying this to sharing the same canonical relational structure and to orbit and age categories. It introduces the orbit category and the age of the induced structure, proving that strong amalgamation is equivalent to an isomorphism between the finite-structure category and the orbit category, and showing that dense subgroups preserve these equivalences. By equating RG-modules with sheaves on a ringed site built from the age and orbit data, it derives Noetherianity results for finitely generated discrete $RG$-modules in the highly homogeneous setting, generalizing prior equivariant Noetherian phenomena. The approach blends topological group theory, model theory (Fraïssé theory), and topos-theoretic methods to obtain structural insights and practical Noetherian consequences for polynomial rings with group symmetries.
Abstract
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.