Endotrivial complexes.

Let G be a finite group, p a prime, and k a field of characteristic p. We introduce the notion of an endotrivial chain complex of p -permutation kG -modules, and study the corresponding group E k (G) of endotrivial complexes. Such complexes are shown to induce splendid Rickard autoequivalences of kG. The elements of E k (G) are determined uniquely by integral invariants arising from the Brauer construction and a degree one character G → k ×. Using ideas from Bouc's theory of biset functors, we provide a canonical decomposition of E k (G) , and as an application, give complete descriptions of E k (G) for abelian groups and p -groups of normal p -rank 1. Taking Lefschetz invariants of endotrivial complexes induces a group homomorphism Λ : E k (G) → O (T (k G)) , where O (T (k G)) is the orthogonal unit group of the trivial source ring. Using recent results of Boltje and Carman, we give a Frobenius stability condition elements in the image of Λ must satisfy. [ABSTRACT FROM AUTHOR]