Suppose that G is a finite group and that k is a field of characteristic p > 0. A kG-module M is an endotrivial module if Homk(M,M) ≅ M∗ ⊗ M ≅ k ⊕ (proj). The endotrivial modules form the Picard group of self-equivalences of the stable category and have been classified for many families of groups. In this lecture I will describe some progress in the classification of endotrivial kG-modules in the case that G is a group of Lie type. We concentrate on the torsion subgroup of the group endotrivial modules, as the torsion free part was determined in earlier work. The torsion part consists mainly of modules whose restrictions to the Sylow subgroup of G are stably trivial. In most cases such modules have dimension 1, but the exceptions are notable.
This is a report on joint work with Jesper Grodal, Nadia Mazza and Dan Nakano.
This video is part of the conference Representation Theory and Geometry that took place at the University of Georgia.
