The weak Lefschetz property, monomial ideals, and lozenges

TL;DR

This paper investigates the weak Lefschetz property and Hilbert functions of certain monomial ideals, establishing conditions for the property and revealing connections to hexagon tilings, with implications for unimodality of Hilbert functions.

Contribution

It demonstrates the weak Lefschetz property for specific monomial ideals in characteristic zero or large characteristic and links these ideals to lozenge tilings, supporting a conjecture.

Findings

Weak Lefschetz property holds for certain monomial ideals in specified characteristics.

Connections established between algebraic properties and hexagon tilings by lozenges.

Hilbert functions are shown to be strictly unimodal for these ideals.

Abstract

We study the weak Lefschetz property and the Hilbert function of level Artinian monomial almost complete intersections in three variables. Several such families are shown to have the weak Lefschetz property if the characteristic of the base field is zero or greater than the maximal degree of any minimal generator of the ideal. Two of the families have an interesting relation to tilings of hexagons by lozenges. This lends further evidence to a conjecture by Migliore, Miro-Roig, and the second author. Finally, using our results about the weak Lefschetz property, we show that the Hilbert function of each level Artinian monomial almost complete intersection in three variables is peaked strictly unimodal.