Monomial bases related to the $n!$ conjecture

Jean Christophe Aval

(Universite de Bordeau)

The $n!$ conjecture, stated by A. Garsia and M. Haiman asserts that the vector space $M_{\mu}$ spanned by all partial derivatives of a polynomial $\Delta_{\mu}$ associated to the partition $\mu$ of $n$ has dimension $n!$.

The aim of this work is to propose a new way to prove the $n!$ conjecture for some particular partitions. The goal 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 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.