Speaker: Andrés R. Vindas Meléndez
Title: Fixed Polytopes of the Permutahedron
Abstract:
In 2010, Stapledon described a generalization of Ehrhart theory with group actions. In 2018, Ardila, Schindler, and I made progress towards answering one of Stapledon’s open problems that asked to determine the equivariant Ehrhart theory of the permutahedron. We proved some general results about the fixed polytopes of the permutahedron, which are the polytopes that are fixed by acting on the permutahedron by a permutation. In particular, we computed their dimension, showed that they are combinatorially equivalent to permutahedra, provided hyperplane and vertex descriptions, and proved that they are zonotopes. Lastly, we obtained a formula for the volume of these fixed polytopes, which is a generalization of Richard Stanley’s result of the volume for the standard permutahedron. Building off of this work, Ardila, Supina, and I determine the equivariant Ehrhart theory of the permutahedron, thereby resolving the open problem. This talk will present background on zonotopes and Ehrhart theory and present some of the aforementioned results.