LEVELNESS OF ORDER POLYTOPES

Since Stanley's [Discrete Comput. Geom., 1 (1986), pp. 9-23] introduction of order polytopes, their geometry has been widely used to examine (algebraic) properties of finite posets. In this paper, we follow this route to examine the levelness property of order polytopes, a property generalizing Gorensteinness. This property has been recently characterized by Miyazaki [J. Algebra, 480 (2017), pp. 215-236] 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 co-NP. Moreover, we show how a necessary condition of levelness of [J. Algebra, 431 (2015), pp. 138-161] can be restated in terms of digraphs. 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.Peer reviewe