New families of Artinian Gorenstein algebras with the weak Lefschetz property
Nasrin Altafi, Rodica Dinu, Shreedevi K. Masuti, Rosa M. Miró-Roig, Alexandra Seceleanu, Nelly Villamizar
TL;DR
This work addresses when graded Artinian Gorenstein algebras with binomial Macaulay dual generators satisfy the weak or strong Lefschetz properties. It develops a gcd-based $\mathrm{WLP}$ criterion and constructs multiple new families of binomial duals $F$ whose associated algebras $A_F$ exhibit $\mathrm{WLP}$ (and sometimes $\mathrm{SLP}$) in arbitrary codimension, using Hessians, connected-sum decompositions, and doubling frameworks. The results extend the codimension-3 success from prior work to higher codimensions and provide concrete tools for building and certifying Lefschetz properties in greater generality. Overall, the paper enhances the structural understanding of Artinian Gorenstein algebras with binomial dual generators and offers practical methods to realize Lefschetz properties in new families.
Abstract
We construct new families of Artinian Gorenstein graded $K$-algebras of arbitrary codimension having binomial Macaulay dual generators and satisfying the weak or the strong Lefschetz property. This is a companion paper to \cite{ADFMMSV}, which studies codimension three algebras having binomial Macaulay dual generators in great depth, establishing in particular that they enjoy the strong Lefschetz property.