  The purpose of this paper is to find a new way to prove the $n!$ conjecture
for particular partitions. The idea is to construct a monomial and explicit
basis for the space $M_{\mu}$. We succeed completely for hook-shaped
partitions, i.e., $\mu=(K+1,1^L)$. We are able to exhibit a basis and to verify
that its cardinality is indeed $n!$, that it is linearly independent and that
it spans $M_{\mu}$. We derive from this study an explicit and simple basis for
$I_{\mu}$, the annihilator ideal of $\Delta_{\mu}$. This method is also
successful for giving directly a basis for the homogeneous subspace of
$M_{\mu}$ consisting of elements of 0 $x$-degree.
