Hilbert Functions of Artinian Gorenstein algebras with the Strong Lefschetz Property

TL;DR

This paper characterizes the Hilbert functions of Artinian Gorenstein algebras with the strong Lefschetz property as SI-sequences, extending previous results and providing new classes of algebras with this property.

Contribution

It proves a complete characterization of Hilbert functions for these algebras and introduces new classes with non-vanishing higher Hessians.

Findings

Hilbert functions of Gorenstein algebras with SLP are exactly SI-sequences.

Extended Harima's characterization from weak to strong Lefschetz property.

Constructed new families of algebras with non-vanishing higher Hessians.

Abstract

We prove that a sequence $h$ of non-negative integers is the Hilbert function of some Artinian Gorenstein algebra with the strong Lefschetz property if and only if it is an SI-sequence. This generalizes the result by T. Harima which characterizes the Hilbert functions of Artinian Gorenstein algebras with the weak Lefschetz property. We also provide classes of Artinian Gorenstein algebras obtained from the ideal of points in $\mathbb{P}^n$ such that some of their higher Hessians have non-vanishing determinants. Consequently, we provide families of such algebras satisfying the SLP.