Endotrivial modules for finite groups via homotopy theory

Abstract:

Classifying endotrivial $kG$-modules, i.e., elements of the Picard group of the stable module category for an arbitrary finite group $G$, has been a long-running quest. By deep work of Dade, Alperin, Carlson, Thévenaz, and others, it has been reduced to understanding the subgroup consisting of modular representations that split as the trivial module $k$ direct sum a projective module when restricted to a Sylow $p$-subgroup. In this paper we identify this subgroup as the first cohomology group of the orbit category on non-trivial $p$-subgroups with values in the units $k^\times$, viewed as a constant coefficient system. We then use homotopical techniques to give a number of formulas for this group in terms of the abelianization of normalizers and centralizers in $G$, in particular verifying the Carlson–Thévenaz conjecture—this reduces the calculation of this group to algorithmic calculations in local group theory rather than representation theory. We also provide strong restrictions on when such representations of dimension greater than one can occur, in terms of the $p$-subgroup complex and $p$-fusion systems. We immediately recover and extend a large number of computational results in the literature, and further illustrate the computational potential by calculating the group in other sample new cases, e.g., for the Monster at all primes.