Representation stability and finite linear groups

TL;DR

This paper develops a new framework extending FI-modules to finite linear groups, establishing structural properties and applying them to prove conjectures and stability theorems in algebraic and geometric contexts.

Contribution

It introduces analogues of FI-modules for finite linear groups and proves their Noetherianity, enabling new proofs of conjectures and stability results in representation theory and topology.

Findings

Proved Noetherianity of the new module categories.

Established homological stability theorems with twisted coefficients.

Provided representation-theoretic proofs of classical stability results.

Abstract

We construct analogues of FI-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as 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.