Incompatible double posets and double order polytopes

TL;DR

This paper generalizes the combinatorial structure of double order polytopes from compatible double posets to all double posets and classifies the 2-level double order polytopes, expanding understanding of their geometric properties.

Contribution

It extends the combinatorial description of double order polytopes to all double posets and provides a classification of 2-level double order polytopes.

Findings

Generalized the structure to all double posets

Classified 2-level double order polytopes

Connected combinatorial and geometric properties

Abstract

In 1986 Stanley associated to a poset the order polytope. The close interplay between its combinatorial and geometric properties makes the order polytope an object of tremendous interest. Double posets were introduced in 2011 by Malvenuto and Reutenauer as a generalization of Stanleys labelled posets. A double poset is a finite set equipped with two partial orders. To a double poset Chappell, Friedl and Sanyal (2017) associated the double order polytope. They determined the combinatorial structure for the class of compatible double posets. In this paper we generalize their description to all double posets and we classify the 2-level double order polytopes.