Abstract: Let G be a reductive algebraic group over an algebraically closed field of characteristic l > 0. To every finite-dimensional rational G-module M, one can associate a (unique minimal) complex of tilting G-modules. The resulting minimal tilting complexes have many favorable properties and they can be used to define so-called generic direct summands in tensor products of simple G-modules. We will discuss the construction of generic direct summands and, time permitting, explain how they can be used to prove a strong necessary condition for the complete reducibility of tensor products of simple G-modules, for G of type An.