Stacks of group representations

TL;DR

This paper introduces a new categorical perspective on group representations, showing that restriction to subgroups can be viewed as an extension-of-scalars and establishing stack structures for various representation categories.

Contribution

It develops a triangulated-categorical framework for understanding subgroup representations as extensions of the whole group, and introduces the sipp topology for stacks of representations.

Findings

Representation restriction is an extension-of-scalars.

Representation categories form stacks on the orbit category.

Sipp topology enables cohomological analysis of endotrivial representations.

Abstract

We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$, the derived and the stable categories of representations of a subgroup $H$ can be constructed out of the corresponding category for $G$ by a purely triangulated-categorical construction, analogous to \'etale extension in algebraic geometry. In the case of finite groups, we then use descent methods to investigate when modular representations of the subgroup $H$ can be extended to $G$. We show that the presheaves of plain, derived and stable representations all form stacks on the category of finite $G$-sets (or the orbit category of $G$), with respect to a suitable Grothendieck topology that we call the sipp topology. When $H$ contains a Sylow subgroup of $G$, we use sipp Cech cohomology to describe the kernel and the image of the homomorphism $T(G)\to T(H)$, where $T(-)$ denotes the group of endotrivial representations.