Minimal Lefschetz decompositions of the derived categories for Grassmannians

We construct two Lefschetz decompositions of the derived category of coherent sheaves on the Grassmannian of k-dimensional subspaces in a vector space of dimension n. Both decompositions admit a Lefschetz basis consisting of equivariant vector bundles. We prove that the first decomposition is full. In the case when n and k are coprime, the decompositions coincide and are minimal. We conjecture that the second decomposition is always full and minimal.