Monomial bases related to the n! conjecture
Abstract
Description
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 $M_μ$. We succeed completely for hook-shaped partitions, i.e., $μ=(K+1,1^L)$. 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 $M_μ$. We derive from this study an explicit and simple basis for $I_μ$, the annihilator ideal of $Δ_μ$. This method is also successful for giving directly a basis for the homogeneous subspace of $M_μ$ consisting of elements of 0 $x$-degree.