  For ample vector bundles $E$ over compact complex varieties $X$ and a Schur
functor $S_I$ corresponding to an arbitrary partition $I$ of the integer $|I|$,
one would like to know the optimal vanishing theorem for the cohomology groups
$H^{p,q}(X, S_I(E))$, depending on the rank of $E$ and the dimension $n$ of
$X$.
  Three years ago (Nov. 1995), in an unpublished paper one of us (W.N.) proved
a vanishing theorem for the situation where the partition $I$ is a hook.
  Here we give a simpler proof of this theorem. We also treat the same problem
under weaker positivity assumptions, in particular under the hypothesis of
ample $\Lambda ^m E$ with $m\in \N^*$. In this case we also need some bound on
the weight $|I|$ of the partition. Moreover, we prove that the same vanishing
condition applies for $H^{q,p}(X, S_I(E))$, with $p,q$ interchanged.
