Monomial bases related to the n! conjecture

TL;DR

This paper constructs explicit monomial bases for the space M_μ related to the n! conjecture, proving the case for hook-shaped partitions and providing new insights into the structure of these algebraic objects.

Contribution

It introduces a new method to explicitly construct bases for M_μ for hook-shaped partitions, advancing understanding of the n! conjecture.

Findings

Successfully constructed bases for M_μ when μ is hook-shaped

Verified the basis has cardinality n! and is linearly independent

Derived explicit bases for the annihilator ideal I_μ

Abstract

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 $M_{\mu}$. We succeed completely for hook-shaped partitions, i.e., $\mu=(K+1,1^L)$. 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 $M_{\mu}$. We derive from this study an explicit and simple basis for $I_{\mu}$, the annihilator 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.