Levelness of Order Polytopes

TL;DR

This paper investigates the levelness property of order polytopes, providing new characterizations, infinite families of level posets, and exploring the levelness of alcoved polytopes and their products.

Contribution

It offers an alternative weighted digraph characterization of levelness, introduces an infinite family of level posets, and analyzes levelness in alcoved polytopes and their products.

Findings

Provided a new characterization of levelness using weighted digraphs.

Constructed an infinite family of level posets.

Showed that the product of two level polytopes may not be level.

Abstract

Since their introduction by Stanley~\cite{StanleyOrderPoly} order polytopes have been intriguing mathematicians as their geometry can be used to examine (algebraic) properties of finite posets. In this paper, we follow this route to examine the levelness property of order polytopes. The levelness property was also introduced by Stanley~\cite{Stanley-CM-complexes} and it generalizes the Gorenstein property. This property has been recently characterized by Miyazaki~\cite{Miyazaki} for the case of order polytopes. We provide an alternative characterization using weighted digraphs. Using this characterization, we give a new infinite family of level posets and show that determining levelness is in $\operatorname{co-NP}$. This family can be used to create infinitely many examples illustrating that the levelness property can not be characterized by the $h^{\ast}$-vector. We then turn to the more general family of alcoved polytopes. We give a characterization for levelness of alcoved polytopes using the Minkowski sum. Then we study several cases when the product of two polytopes is level. In particular, we provide an example where the product of two level polytopes is not level.