Monotone Paths on Cross-Polytopes

TL;DR

This paper studies the structure of monotone path polytopes for cross-polytopes, revealing their face lattice is isomorphic to the lattice of intervals in the sign poset, and explores their combinatorial properties.

Contribution

It provides a detailed analysis of the monotone path polytope of cross-polytopes, including its face lattice, $f$-vector, realizations, and facets, extending the theory beyond simplices and hyper-cubes.

Findings

Face lattice is isomorphic to the lattice of intervals in the sign poset.

Analyzes the $f$-vector, realizations, and facets of the MPP for cross-polytopes.

Extends the understanding of monotone path polytopes to a new class of polytopes.

Abstract

In the early 1990's, Billera and Sturmfels introduced the monotone path polytope (MPP), a special case of the general theory of fiber polytopes that associates a polytope to a pair $(P,\varphi)$ of a polytope $P$ and linear functional $\varphi$. In that same paper, they showed that MPPs of simplices and hyper-cubes are combinatorial cubes and permutahedra respectively. Their work has lead to many developments in combinatorics. Here we investigate the monotone paths for generic orientations of cross-polytopes. We show the face lattice of its MPP is isomorphic to the lattice of intervals in the sign poset from oriented matroid theory. We look at its $f$-vector, its realizations, and facets.