Classify families of Artinian k-algebras with the strong Lefschetz property

Determine for which families of standard graded Artinian k-algebras A = R/I over a field k of characteristic zero there exists a linear form ℓ ∈ A1 such that, for all integers t ≥ 1 and all degrees i, the multiplication maps ×ℓ^t: A_i → A_{i+t} have maximal rank (injective or surjective), i.e., for which A has the strong Lefschetz property.

Background

The strong Lefschetz property (SLP) for a graded Artinian k-algebra A = R/I requires the existence of a linear form ℓ ∈ A1 such that the multiplication maps ×ℓ^t: A_i → A_{i+t} have maximal rank for all t ≥ 1. The weak Lefschetz property (WLP) is the special case t = 1.

A foundational result by Stanley and Watanabe shows that Artinian monomial complete intersections have the SLP. Motivated by this, the paper investigates two specific families of monomial almost complete intersections and provides complete criteria for SLP in those cases: when the extra generator has support in two variables, and when the Hilbert series is symmetric. Despite these advances, the broader classification across all families of Artinian algebras remains unresolved, as highlighted by the authors’ explicit statement of this overarching open problem.