Representation stability and finite linear groups

Abstract

We study analogues of $\mathtt{FI}$-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings, and we prove basic structural properties such as local Noetherianity. Applications include a proof of the Lannes–Schwartz Artinian conjecture in the generic representation theory of finite fields, very general homological stability theorems with twisted coefficients for the general linear and symplectic groups over finite rings, and representation-theoretic versions of homological stability for congruence subgroups of the general linear group, the automorphism group of a free group, the symplectic group, and the mapping class group.