  Starting point of the present work is a conjecture of F. Catanese which says
that in the derived category of coherent sheaves on any rational homogeneous
manifold G/P there should exist a complete strong exceptional poset and a
bijection of the elements of the poset with the Schubert varieties in G/P such
that the partial order on the poset is the order induced by the
Bruhat-Chevalley order. The goal of this work is to provide further evidence
for Catanese's conjecture, clarify some aspects of it and supply new
techniques. In particular we prove a theorem on the derived categories of
quadric bundles, and show how one can find "small" generating sets for D^b(X)
on symplectic or orthogonal isotropic Grassmannians by fibrational techniques.-
The last section discusses a different approach based on a theorem of M. Brion
and cellular resolutions of monomial ideals.
