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

Federico Ardila (San Francisco State University / Universidad de Los Andes)

Live Stream MPI für Mathematik in den Naturwissenschaften Leipzig (Live Stream)

Abstract

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 $(2 n - 2)$ -dimensional polytopes with $3^{n} - 3$ facets arose in our investigation of conormal fans: the bipermutohedron and the harmonic polytope. The bipermutohedron has $(2 n)! / 2^{n}$ 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 $(n!)^{2} (1 + 1 / 2 + \dots + 1 / n)$ 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.

Links

conference

06.04.21 09.04.21

(Polytop)ics: Recent advances on polytopes (Polytop)ics: Recent advances on polytopes

MPI für Mathematik in den Naturwissenschaften Leipzig Live Stream

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Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Federico Castillo

Max Planck Institute for Mathematics in the Sciences

Giulia Codenotti

Goethe University Frankfurt

Benjamin Schröter

Royal Institute of Technology (KTH)