The harmonic polytope

TL;DR

This paper investigates the harmonic polytope's combinatorial structure, providing formulas for its vertices, facets, and volume, linking it to matroid theory, toric ideals, and generalized permutahedra.

Contribution

It offers a detailed combinatorial and volumetric analysis of the harmonic polytope, connecting it to algebraic and geometric structures in matroid and toric geometry.

Findings

The harmonic polytope is a (2n-2)-dimensional polytope.

It has $(n!)^2(1+rac12+rac1n)$ vertices and $3^n-3$ facets.

Its volume is a weighted sum related to toric ideals and lattice points.

Abstract

We study the harmonic polytope, which arose in Ardila, Denham, and Huh's work on the Lagrangian geometry of matroids. We describe its combinatorial structure, showing that it is a $(2n-2)$-dimensional polytope with $(n!)^2(1+\frac12+\cdots+\frac1n)$ vertices and $3^n-3$ facets. We also give a formula for its volume: it is a weighted sum of the degrees of the projective varieties of all the toric ideals of connected bipartite graphs with $n$ edges; or equivalently, a weighted sum of the lattice point counts of all the corresponding trimmed generalized permutahedra.