Representation stability and outer automorphism groups

TL;DR

This paper explores the stability properties of representations of outer automorphism groups across various finite groups, generalizing existing theories and connecting to rational global homotopy invariants.

Contribution

It introduces a new abelian category framework for these representations and establishes conditions for stability and noetherian properties, extending prior stability results.

Findings

The abelian category U is locally noetherian under certain conditions.

Analogues of central and representation stability are proved in this new setting.

Global homotopy invariants are shown to exhibit representation stability.

Abstract

In this paper we study families of representations of the outer automorphism groups indexed on a collection of finite groups $\mathcal{U}$. We encode this large amount of data into a convenient abelian category $\mathcal{A}\mathcal{U}$ which generalizes the category of VI-modules appearing in the representation theory of the finite general linear groups. Inspired by work of Church--Ellenberg--Farb, we investigate for which choices of $\mathcal{U}$ the abelian category is locally noetherian and deduce analogues of central stability and representation stability results in this setting. Finally, we show that some invariants coming from rational global homotopy theory exhibit representation stability.