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, which by deep work of Dade, Alperin, Carlson, Thévenaz, and others, has been reduced to understanding the subgroup consisting of modular representations that split as the trivial module 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 × , 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.

Higher limits over the fusion orbit category

“…We first recall some definitions on categories with group action. We follow the terminology and notation introduced by Grodal in [13] and [14]. Let G be a discrete group.…”

Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup

“…3. The third main ingredient relies on major new results obtained by J. Grodal through the use of homotopy theory in [Gro99], or more precisely on a slight extension of the main theorem of [Gro99] recently obtained by D. Craven in [Cra20] and which provides us with purely group-theoretic techniques to deal with Grodal's description of K(G) in [Gro99, Theorem 4.27]. The latter results will in particular enable us to treat families of groups related to the finite groups of Lie type SL 3 (q) with q ≡ 3 (mod 4) and SU 3 (q) with q ≡ 1 (mod 4).…”

The reductive Borel-Serre compactification as a model for unstable algebraic K-theory

“…The simplest non-example is a finite field, and our third theorem analyses this case to see the difference with the plus construction. As we will explain below, the resulting theorem should properly be attributed to Jesper Grodal, since in [Gro18] he proved a more general result in the context of arbitrary finite groups which specialises to this result for G = GL(M). However, we do give an independent proof based on the general machinery for analysing RBS(M) categories that we develop.…”

“…The crucial point is part 2, that | RBS(V )| has the F p -homology of a point. This can also be deduced from a more general theorem of Jesper Grodal, [Gro18]. Indeed, Grodal's Theorem 4.3 says that for any finite group G and prime p, if C denotes the p-radical orbit category of G, then |C| has the F p -homology of a point.…”