Lefschetz properties of balanced 3-polytopes

TL;DR

This paper investigates Lefschetz properties of Artinian reductions of Stanley-Reisner rings of balanced 3-polytopes, establishing conditions under which the strong Lefschetz property holds, with combinatorial characterizations for specific balanced polytopes.

Contribution

It proves the strong Lefschetz property for Artinian reductions of Stanley-Reisner rings of balanced 3-polytopes under certain conditions and characterizes special balanced polytopes with combinatorial criteria.

Findings

Strong Lefschetz property holds for balanced 3-polytopes in characteristic not 2 or 3.

Characterization of (2,1)-balanced polytopes via Laman-type conditions.

Special systems of parameters induced by coloring are key to the properties.

Abstract

In this paper, we study Lefschetz properties of Artinian reductions of Stanley-Reisner rings of balanced simplicial $3$-polytopes. A $(d-1)$-dimensional simplicial complex is said to be balanced if its graph is $d$-colorable. If a simplicial complex is balanced, then its Stanley-Reisner ring has a special system of parameters induced by the coloring. We prove that the Artinian reduction of the Stanley-Reisner ring of a balanced simplicial $3$-polytope with respect to this special system of parameters has the strong Lefschetz property if the characteristic of the base field is not two or three. Moreover, we characterize $(2,1)$-balanced simplicial polytopes, i.e., polytopes with exactly one red vertex and two blue vertices in each facet, such that an analogous property holds. In fact, we show that this is the case if and only if the induced graph on the blue vertices satisfies a Laman-type combinatorial condition.