Artinian Gorenstein algebras with binomial Macaulay dual generator

This paper initiates a systematic study for key properties of Artinian Gorenstein \ (K\) -algebras having binomial Macaulay dual generators. In codimension 3, we demonstrate that all such algebras satisfy the strong Lefschetz property, can be constructed as a doubling of an appropriate 0-dimensional scheme in \ (P²\), and we provide an explicit characterization of when they form a complete intersection. For arbitrary codimension, we establish sufficient conditions under which the weak Lefschetz property holds and show that these conditions are optimal.