Monomial ideals and the failure of the Strong Lefschetz property

Abstract

We give a sharp lower bound for the Hilbert function in degree $d$ of artinian quotients $\mathbb{k}[x_1,\dots ,x_n]/I$ failing the Strong Lefschetz property, where $I$ is a monomial ideal generated in degree $d\ge 2$. We also provide sharp lower bounds for other classes of ideals, and connect our result to the classification of the Hilbert functions forcing the Strong Lefschetz property by Zanello and Zylinski.