A Continuous Family of Marked Poset Polytopes

TL;DR

This paper introduces a continuous family of polytopes generalizing marked order and chain polytopes, parametrized by a hypercube, with explicit vertex descriptions and Ehrhart equivalence.

Contribution

It defines a new family of polytopes for marked posets, establishes transfer maps for Ehrhart equivalence, and describes vertices using tropical hyperplane arrangements.

Findings

Vertices are parametrized by the hypercube's faces.

The combinatorial type remains constant within each face.

Explicit vertex descriptions are provided for generic parameters.

Abstract

For any marked poset we define a continuous family of polytopes, parametrized by a hypercube, generalizing the notions of marked order and marked chain polytopes. By providing transfer maps, we show that the vertices of the hypercube parametrize an Ehrhart equivalent family of lattice polytopes. The combinatorial type of the polytopes is constant when the parameters vary in the relative interior of each face of the hypercube. Moreover, with the help of a subdivision arising from a tropical hyperplane arrangement associated to the marked poset, we give an explicit description of the vertices of the polytope for generic parameters.