# A note on Artin Gorenstein algebras with Hilbert function (1,4,k,k,4,1)

## TL;DR

This paper investigates Artin Gorenstein algebras with specific Hilbert functions, establishing conditions under which their Betti tables are uniquely determined and linking the Strong and Weak Lefschetz properties.

## Contribution

It proves that all such algebras with the given Hilbert function have the Strong Lefschetz property if they have the Weak Lefschetz property, and shows when Betti tables are uniquely determined.

## Key findings

- All Artin Gorenstein algebras with Hilbert function (1,4,k,k,4,1) have the Strong Lefschetz property if they have the Weak Lefschetz property.
- For k=4, the Hilbert function determines the Betti table uniquely.
- Complete intersection or equigenerated ideals lead to a unique Betti table.

## Abstract

We study the free resolutions of some Artin Gorenstein algebras of Hilbert function $(1,4,k,k,4,1)$ and we prove that all such algebras have the Strong Lefschetz property if they have the Weak Lefschetz property. In the case $k=4$ we prove that the Hilbert function alone fixes the betti table. For higher $k$ stronger conditions on the algebras are needed to fix the betti table. In particular, if the algebra is a complete intersection or if it is defined by an equigenerated ideal then the betti table is unique.