On Artinian Gorenstein algebras associated to the face posets of regular polyhedra

We introduce Artinian Gorenstein algebras defined by the face posets of regular polyhedra. We consider the strong Lefschetz property and Hodge–Riemann relation for the algebras. We show the strong Lefschetz property of the algebras for all Platonic solids. On the other hand, for some Platonic solids, we show that the algebras do not satisfy the Hodge–Riemann relation with respect to some strong Lefschetz elements.