Levelness of Order Polytopes

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.