Endotrivial modules for finite groups via homotopy theory

TL;DR

This paper uses homotopy theory to classify endotrivial modules for finite groups, providing formulas and computational methods that simplify the process and verify longstanding conjectures, including calculations for complex groups like the Monster.

Contribution

It identifies the subgroup of endotrivial modules as a first cohomology group and develops homotopical formulas to compute it, confirming the Carlson-Thevenaz conjecture.

Findings

Verified the Carlson-Thevenaz conjecture.

Provided formulas linking cohomology to normalizers and centralizers.

Calculated endotrivial groups for the Monster group at all primes.

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, Thevenaz, 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^x, 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-Thevenaz 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.