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A tale of two polytopes 1: The bipermutohedron.

Federico Ardila

In our work on the Lagrangian geometry of matroids, we introduced the conormal fan of a matroid $M$. We used it to reinterpret the Chern-Schwartz-MacPherson cycle of $M$ geometrically, and to prove Brylawski and Dawson's conjectures on the log-concavity of the $h$-vector of $M$. Two related polytopes arose in our investigation of conormal fans: the bipermutohedron and the harmonic polytope. This talk will discuss the combinatorics of the bipermutohedron, a $(2n-2)$-dimensional polytope with $(2n)!/2^n$ vertices, $3^n-3$ facets, and an elegant face structure. In particular we compute its $h$-polynomial, which we call the biEulerian polynomial, and we prove that it is real-rooted. This talk will present joint work with Graham Denham and June Huh. I will try to make it as self-contained as possible. The two parts of this tale of two polytopes can be followed independently.