Classification of Perazzo algebras with the weak Lefschetz property

Classify all artinian Perazzo algebras that have the weak Lefschetz property, including criteria determining the property from their algebraic or Hilbert-series data.

Background

Perazzo algebras arise through Macaulay inverse systems from Perazzo forms, and the paper constructs a large family of G-quadratic Gorenstein Perazzo algebras that fail the weak Lefschetz property. Existing results determine the presence of the weak Lefschetz property from the Hilbert series when the number of variables is small, but do not provide a complete classification.

The authors explicitly identify the full classification of Perazzo algebras with the weak Lefschetz property as unresolved. This problem is broader than the paper’s negative constructions and concerns characterizing precisely which Perazzo algebras possess the property.