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# A Tale of Two Polytopes: the Bipermutahedron and the Harmonic Polytope

## Blackwell Tapia Conference 2021 November 18, 2021 - November 20, 2021

November 19, 2021 (12:00 PM PST - 12:50 PM PST)
Speaker(s): Federico Ardila (San Francisco State University)
Location: MSRI: Simons Auditorium
Tags/Keywords
• Polytopes

• f-vector

• h-vector

• Ehrhart theory

• toric ideal

• generalized permutahedron

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

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

Abstract

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.

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 self-contained as possible, and will feature joint work with Graham Denham, Laura Escobar, and June Huh.

Supplements
 Tale of Two Polytopes: the Bipermutahedron and the Harmonic Polytope 3.21 MB application/pdf Download
Video/Audio Files

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

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