LEVELNESS OF ORDER POLYTOPES

Christian Haase*, Florian Kohl, Akiyoshi Tsuchiya

*Corresponding author for this work

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Abstract

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.

Original languageEnglish
Pages (from-to)1261-1280
Number of pages20
JournalSIAM Journal on Discrete Mathematics
Volume34
Issue number2
DOIs
Publication statusPublished - 2020
MoE publication typeA1 Journal article-refereed

Keywords

  • order polytopes
  • Bellman-Ford algorithm
  • posets
  • level algebras
  • alcoved polytopes
  • PROPERTY
  • RINGS

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