A Tale of Two Polytopes: the Bipermutahedron and the Harmonic Polytope
Blackwell Tapia Conference 2021 November 18, 2021  November 20, 2021
Location: SLMath: Eisenbud Auditorium
Polytopes
fvector
hvector
Ehrhart theory
toric ideal
generalized permutahedron
49K21  Optimality conditions for problems involving relations other than differential equations
49K45  Optimality conditions for problems involving randomness [See also 93E20]
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 fvector, which counts the faces of each dimension, is given by a simple evaluation of the three variable RogersRamanujan function. Its hvector, which gives the dimensions of the intersection cohomology of the associated toric variety, is logconcave.
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.
These two polytopes are related by a surprising fact: in any dimension, the Minkowski quotient of the bipermutahedron and the harmonic polytope equals 2.
The talk will be as selfcontained as possible, and will feature joint work with Graham Denham, Laura Escobar, and June Huh.
Tale of Two Polytopes: the Bipermutahedron and the Harmonic Polytope

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A Tale Of Two Polytopes: The Bipermutahedron And The Harmonic Polytope
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