The Equivariant Ehrhart Theory of the Permutahedron

event Friday, November 13, 2020
access_time 3:30 PM (CT)
room Zoom
free_breakfast Tea at 4:30 PM (CT) in Zoom

Abstract: In 2010, motivated by representation theory, 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 the work of the aforementioned, Ardila, Supina, and I determine the equivariant Ehrhart theory of the permutahedron, thereby resolving the open problem. This project presents formulas for the Ehrhart quasipolynomials and Ehrhart Series of the fixed polytopes of the permutahedron, along with other results regarding interpretations of the equivariant analogue of the Ehrhart series. This is joint work with Federico Ardila (San Francisco State University, Anna Schindler (North Seattle College) and Mariel Supina (UC Berkeley).

For more information on this event, please contact Emily Gunawan.