  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 of them admit a Lefschetz basis consisting of equivariant
vector bundles. We prove fullness of the first decomposition and conjecture it
for the second one. In the case when $n$ and $k$ are coprime these
decompositions coincide and are minimal. In general, we conjecture minimality
of the second decomposition.
