MIT-Harvard-MSR Combinatorics Seminar
Wednesday, September 22, 2021 at 4:00pm to 5:00pm
(NOTE: The talk will start at 4PM sharp,)
Speaker: Federico Ardila (San Francisco State University)
Title: A tale of two polytopes: The bipermutahedron and the harmonic polytope
The harmonic polytope and the bipermutahedron are two related polytopes which arose in our work with Graham Denham and June Huh on the Lagrangian geometry of matroids. This talk will briefly explain their geometric origin and discuss their algebraic and geometric combinatorics.
The bipermutahedron is a (2n−2)-dimensional polytope with (2n!)/2^n vertices and 3^n−3 facets. Its f-vector, which counts the faces of each dimension, is given by a simple evaluation of the three variable Rogers-Ramanujan function. Its h-vector, which gives the dimensions of the intersection cohomology of the associated toric variety, is log-concave.
The harmonic polytope is a (2n−2)-dimensional polytope with (n!)^2(1+1/2+...+1/n) vertices and 3^n−3 facets. Its volume is a weighted sum of the degrees of the projective varieties of all the toric ideals of connected bipartite graphs with n edges.
The talk will be as self-contained as possible, and will feature joint work with Graham Denham, Laura Escobar, and June Huh.