Endotrivial representations of finite groups and equivariant line bundles on the Brown complex

Abstract

We relate endotrivial representations of a finite group in characteristic p to equivariant line bundles on the simplicial complex of non-trivial p-subgroups, by means of weak homomorphisms.Dedicated to Serge Bouc on the occasion of his 60 th birthday IntroductionLet G be a finite group, p a prime dividing the order of G and k a field of characteristic p. For the whole paper, we fix a Sylow p-subgroup P of G.Consider the endotrivial kG-modules M , i.e. those finite dimensional k-linear representations M of G which are ⊗-invertible in the stable category kG -stab = kG -mod / kG -proj; this means that the kG-module End k (M ) is isomorphic to the trivial module k plus projective summands. The stable isomorphism classes of these endotrivial modules form an abelian group, T k (G), under tensor product. This important invariant has been fully described for p-groups in celebrated work of Carlson and Thévenaz [CT04, CT05]. Therefore, for general finite groups G, the focus has moved towards studying the relative version: T k (G, P ) := Ker T k (G) → T k (P ) .

Endotrivial modules for finite groups via homotopy theory

“…Before moving to homology decompositions we record the following corollary of Theorem A, by (1.10) and (1.6) above: From this perspective, exotic Sylow-trivial modules parametrize the failure of the collection of nontrivial p-subgroups to be 'H 1 (−; Z)-ample' in the spirit of Dwyer [Dwy97,1.3]; see Remark 4.7 and Theorem 4.34. The corollary also lets us deduce a very recent result of Balmer [Bal18]; see Remark 4.8. 1.2.…”