Representation stability and outer automorphism groups

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 th...

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Bibliographic Details
Published in:arXiv.org 2021-05
Main Authors: Pol, Luca, Strickland, Neil P
Format: Article
Language:English
Subjects:
Automorphisms
Homotopy theory
Stability
Online Access:Get full text
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Summary: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.
ISSN:2331-8422