Artinian Gorenstein algebras of socle degree three which have the weak Lefschetz property

Let k be an arbitrary field and Φ be the Macaulay inverse system for a standard graded Artinian Gorenstein k-algebra A of arbitrary embedding dimension d and socle degree three. Assume that A has the weak Lefschetz property. We identify generators for the defining ideal of A as a quotient of a polynomial ring P over k with d variables and we give an explicit homogeneous resolution, $X$, of A by free P-modules. We identify a symmetric bilinear form G which determines how to turn $X$ into the minimal resolution of A. In particular, when G is identically zero, then $X$ is already the minimal resolution of A.

The resolution $X$ is closely related to the resolution of a Gorenstein algebra with socle degree two. A Gorenstein algebra with socle degree two has a resolution that is as linear as possible.

The corresponding project has previously been carried out (by the present author, and also by Macias Marques, Veliche, and Weyman), when the embedding dimension d is equal to 4.