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

TL;DR

This paper introduces Artinian Gorenstein algebras based on face posets of regular polyhedra, demonstrating the strong Lefschetz property for all Platonic solids but identifying cases where the Hodge--Riemann relation fails.

Contribution

It establishes the strong Lefschetz property for these algebras associated with all Platonic solids and explores their Hodge--Riemann relations, revealing nuanced differences.

Findings

Strong Lefschetz property holds for all Platonic solids.

Hodge--Riemann relation fails for some Platonic solids.

Provides new algebraic insights into face posets of regular polyhedra.

Abstract

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.