The strong Lefschetz property for quadratic reverse lexicographic ideals

Abstract

Consider ideals $I$ of the form \[ I=(x_1^2,\dots, x_n^2)+\mathrm{RLex}(x_ix_j) \] where $\mathrm{RLex}(x_ix_j)$ is the ideal generated by all the square-free monomials which are greater than or equal to $x_ix_j$ in the reverse lexicographic order. We will determine some interesting properties regarding the shape of the Hilbert series of $I$. Using a theorem of Lindsey, this allows for a short proof that any algebra defined by $I$ has the strong Lefschetz property when the underlying field is of characteristic zero. Building on recent work by Phuong and Tran, this result is then extended to fields of sufficiently high positive characteristic. As a consequence, this shows that for any possible number of minimal generators for an artinian quadratic ideal there exists such an ideal minimally generated by that many monomials and defining an algebra with the strong Lefschetz property.