The harmonic polytope

Abstract

We study the harmonic polytope, which arose in Ardila, Denham, and Huh's work on the Lagrangian geometry of matroids. We show that it is a (2n − 2)-dimensional polytope withvertices and 3 n − 3 facets. We 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.

The bipermutahedron

“…Proof. Again, we already know this statement must be true because the normal fan of the harmonic polytope is a coarsening of the bipermutahedral fan [4], so its support function H must be in the nef cone of Σ n,n . However, giving a direct proof will allow us to derive a stronger result.…”

“…, a 8 ) = (3, 1, 1, 2, 2, 2, 1, 2). Then As shown in [4], the harmonic polytope H n,n is given by the following minimal inequality description: We translate it by the vector −( n+1 2 + 1 n )(e E + f E ) so that it lands on the subspace M n × M n given by e∈ [n] x e = e∈[n] y e = 0. Then we have the following statement.…”

“…However, giving a direct proof will allow us to derive a stronger result. 10 Acknowledgments I would like to extend my warm gratitude to Graham Denham and June Huh for the very rewarding collaboration [3] that led to the construction of the harmonic polytope and the bipermutahedron, and to Laura Escobar [4] for the very rewarding collaboration that taught me most of what I know about the harmonic polytope. Several of the results in this paper were inspired by conversations with them.…”

“…The harmonic polytope H n,n is studied in our paper [4] with Laura Escobar. The bipermutahedron Π n,n is the main object of study of this paper.…”