UC Riverside

The category $\FI_G$ has been studied in relation to the representation theory of the wreath products $G \wr S_n$, where $G$ is a finite group. We take $G=A$ to be abelian and define the category $\FI_A^H$ associated to certain subgroups $J_n$ of $A \wr S_n$, including the finite complex reflection groups $G(m,p,n)$ defined by Shepard and Todd. We also give axioms for ``$\FI$-like'' categories $\C$, and prove the equivalence of the noetherian property for $\C$-modules over $k$ with the noetherian property for $\A$-modules over $k$, where $\A$ is a suitably restricted subcategory of $\C$ and $k$ is a commutative noetherian ring with unity. We apply this result to show that representation stability for a sequence of finite-dimensional $J_n$-representations over $\mathbb{C}$ is equivalent to finite generation of the corresponding $\FI_A^H$-module. We also prove the analogous result for representations of the alternating groups. Lastly, we prove homological stability for the family $\{J_n\}$ with twisted coefficients from a finitely generated $\FI_A^H$-module over $\Z$, as well as the equivalence of certain Serre quotient categories of locally noetherian $\C$- and $\A$-modules.