The harmonic polytope

Federico Ardila ^[1] ; Laura Escobar ^[2]
1. [1] San Francisco State University, USA; Universidad de Los Andes, Colombia
2. [2] Washington University, USA
Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 27, Nº. 5, 2021
Idioma: inglés
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Resumen
- 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+12+⋯+1n) vertices and 3n−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.