Realizing the $s$-permutahedron via flow polytopes

TL;DR

This paper confirms a conjecture by providing three geometric realizations of the $s$-permutahedron for strict compositions, connecting flow polytopes, hypercube subdivisions, and tropical geometry.

Contribution

It offers the first explicit geometric realizations of the $s$-permutahedron for strict compositions, using flow polytopes, Cayley trick, and tropical geometry.

Findings

Three geometric realizations of the $s$-permutahedron are constructed.

The realizations include dual graphs of flow polytope triangulations, hypercube subdivisions, and tropical geometry complexes.

Vertices of the polyhedral complex are explicitly described, supporting the conjecture.

Abstract

Ceballos and Pons introduced the $s$-weak order on $s$-decreasing trees, for any weak composition $s$. They proved that it has a lattice structure and further conjectured that it can be realized as the $1$-skeleton of a polyhedral subdivision of a polytope. We answer their conjecture in the case where $s$ is a strict composition by providing three geometric realizations of the $s$-permutahedron. The first one is the dual graph of a triangulation of a flow polytope of high dimension. The second one, obtained using the Cayley trick, is the dual graph of a fine mixed subdivision of a sum of hypercubes that has the conjectured dimension. The third one, obtained using tropical geometry, is the $1$-skeleton of a polyhedral complex for which we can provide explicit coordinates of the vertices and whose support is a permutahedron as conjectured.