  The aim of this work is to construct a monomial and explicit basis for the
space $M_{\mu}$ relative to the $n!$ conjecture. We succeed completely for
hook-shaped partitions, i.e. $\mu=(K+1,1^L)$. We are indeed able to exhibit a
basis and to verify that its cardinality is $n!$, that it is linearly
independent and that it spans $M_{\mu}$. We deduce from this study an explicit
and simple basis for $I_{\mu}$, the annulator 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.
