Bases explicites et conjecture n!

Abstract

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.