Bases explicites et conjecture n!

TL;DR

This paper constructs explicit bases for certain algebraic spaces related to the $n!$ conjecture, confirming the conjecture for hook-shaped partitions and providing tools for understanding their structure.

Contribution

It provides a complete explicit basis for $M_{rac{1}{2}}$ spaces for hook-shaped partitions, verifying the $n!$ conjecture in this case.

Findings

Established a basis with cardinality $n!$ for $M_{rac{1}{2}}$ for hook partitions

Verified linear independence and spanning properties of the basis

Derived explicit bases for the annihilator ideal $I_{rac{1}{2}}$

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.