Principal Radical Systems, Lefschetz Properties and Perfection of Specht Ideals of Two-Rowed Partitions

TL;DR

This paper proves the perfection of Specht ideals for two-row partitions over fields with certain characteristics, establishes related properties, and connects these to the weak Lefschetz property, providing new insights into Cohen-Macaulayness.

Contribution

It demonstrates the perfection of Specht ideals in broader settings, links them to Lefschetz properties, and offers a new proof of Cohen-Macaulayness for specific algebraic sets.

Findings

Specht ideal of two-row partitions is perfect over fields with characteristic zero or large enough

Perfection and properties of variants of Specht ideals are established

Connection between Specht ideals and the weak Lefschetz property is demonstrated

Abstract

We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results in particular give a self-contained proof of Cohen-Macaulayness of certain $h$-equals sets, a result previously obtained by Etingof-Gorsky-Losev over the complex numbers using rational Cherednik algebras.