A tale of two polytopes: The bipermutohedron and the harmonic polytope

In our work with Graham Denham and June Huh on the Lagrangian geometry of matroids, we introduced the conormal fan of a matroid M. Using tools from combinatorial Hodge theory, we used this fan to prove Brylawski and Dawson's conjectures on the log-concavity of the h-vector of the broken circuit and the independence complex of M.

Two $(2n-2)$-dimensional polytopes with $3^n-3$ facets arose in our investigation of conormal fans: the bipermutohedron and the harmonic polytope. The bipermutohedron has $(2n)!/2^n$ vertices. Its h-polynomial is an analog of the Eulerian polynomial. It is real-rooted, and hence its coefficients are log-concave and unimodal. The harmonic polytope has $(n!)^2(1+1/2+…+1/n)$ vertices. Its volume is a combination of the degrees of the projective varieties of all the toric ideals of connected bipartite graphs with n edges.

My talk will discuss these combinatorial constructions. The work on the harmonic polytope is joint with Laura Escobar.