Stacks of group 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 extensionofscalars. 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 triangulatedcategorical construction, analogous to étale 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2013
 DOI:
 10.48550/arXiv.1302.6290
 arXiv:
 arXiv:1302.6290
 Bibcode:
 2013arXiv1302.6290B
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Category Theory;
 20C20;
 14A20;
 18F10;
 18E30;
 18C20
 EPrint:
 Slightly revised version of the 2012 June 21 version