Representation stability and outer automorphism groups

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 which generalizes the category of VI-modules appearing in the representation theory of the finite general linear groups. Inspired by work of T. Church et al. [Duke Math. J. 164, No. 9, 1833–1910 (2015; Zbl 1339.55004)], 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.