Gr\"obner bases of radical Li-Li type ideals associated with partitions

TL;DR

This paper introduces a new class of ideals generalizing Specht and radical Li-Li ideals, analyzing their radicalness and Gr"obner bases to advance understanding of algebraic structures linked to partitions.

Contribution

It defines a new class of ideals that unify Specht and radical Li-Li ideals, and investigates their radicalness and Gr"obner bases.

Findings

Established conditions for radicalness of the new ideals.

Derived explicit Gr"obner bases for the class of ideals.

Unified framework for Specht and radical Li-Li ideals.

Abstract

For a partition $\lambda$ of $n$, the _Specht ideal_ $I_\lambda \subset K[x_1, \ldots, x_n]$ is the ideal generated by all Specht polynomials of shape $\lambda$. In their unpublished manuscript, Haiman and Woo showed that $I_\lambda$ is a radical ideal, and gave its universal Gr\"obner bases (recently, Murai et al. published a quick proof of this result). On the other hand, an old paper of Li and Li studied analogous ideals, while their ideals are not always radical. In the present paper, we introduce a class of ideals which generalizes both Specht ideals and _radical_ Li-Li ideals, and study their radicalness and Gr\"obner bases.