The weak Lefschetz property of equigenerated monomial ideals

TL;DR

This paper establishes a precise lower bound for the Hilbert function of monomial algebras that do not satisfy the weak Lefschetz property, focusing on ideals invariant under cyclic group actions in polynomial rings.

Contribution

It provides a sharp lower bound for the Hilbert function and classifies invariant ideals based on the weak Lefschetz property for any number of variables and degrees.

Findings

Determined a sharp lower bound for the Hilbert function in failing cases.

Classified invariant ideals according to the weak Lefschetz property.

Extended results to all degrees and variable counts with cyclic group actions.

Abstract

We determine a sharp lower bound for the Hilbert function in degree $d$ of a monomial algebra failing the weak Lefschetz property over a polynomial ring with $n$ variables and generated in degree $d$, for any $d\geq 2$ and $n\geq 3$. We consider artinian ideals in the polynomial ring with $n$ variables generated by homogeneous polynomials of degree $d$ invariant under an action of the cyclic group $\mathbb{Z}/d\mathbb{Z}$, for any $n\geq 3$ and any $d\geq 2$. We give a complete classification of such ideals in terms of the weak Lefschetz property depending on the action.