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 \((n!)^2\left( 1+\frac{1}{2}+\cdots +\frac{1}{n}\right) \) 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.
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Notes
Of course, this is the same information as the relative order of the \(z_i\)s. We use the reverse order because it is consistent with our choice of working with inner normal fans.
This should not be confused with the permutohedron, which makes no further appearances in the paper.
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Acknowledgements
The first author would like to thank Graham Denham and June Huh for the very rewarding collaboration that led to the construction of the harmonic polytope. We would like to thank the anonymous referees for their careful reading of the work and their valuable suggestions to improve the exposition.
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F. Ardila: Partially supported by NSF Grant DMS-1855610 and Simons Fellowship 613384. L. Escobar: Partially supported by NSF Grant DMS-1855598.
