Author ORCID Identifier
Date Available
5-24-2021
Year of Publication
2021
Degree Name
Doctor of Philosophy (PhD)
Document Type
Doctoral Dissertation
College
Arts and Sciences
Department/School/Program
Mathematics
First Advisor
Dr. Benjamin Braun
Abstract
The first part of this dissertation deals with the equivariant Ehrhart theory of the permutahedron. As a starting point to determining the equivariant Ehrhart theory of the permutahedron, Ardila, Schindler, and I obtain a volume formula for the rational polytopes that are fixed by acting on the permutahedron by a permutation, which generalizes a result of Stanley’s for the volume for the standard permutahedron. Building from the aforementioned work, Ardila, Supina, and I determine the equivariant Ehrhart theory of the permutahedron, thereby resolving an open problem posed by Stapledon. We provide combinatorial descriptions of the Ehrhart quasipolynomial and Ehrhart series of the fixed polytopes of the permutahedron. Additionally, we answer questions regarding the polynomiality of the equivariant analogue of the h*-polynomial.
The second part of this dissertation deals with decompositions of the h*-polynomial for rational polytopes. An open problem in Ehrhart theory is to classify all Ehrhart quasipolynomials. Toward this classification problem, one may ask for necessary in- equalities among the coefficients of an h*-polynomial. Beck, Braun, and I contribute such inequalities when P is a rational polytope. Additionally, we provide two decompositions of the h*-polynomial for rational polytopes, thereby generalizing results of Betke and McMullen and Stapledon. We use our rational Betke–McMullen formula to provide a novel proof of Stanley’s Monotonicity Theorem for rational polytopes.
Digital Object Identifier (DOI)
https://doi.org/10.13023/etd.2021.225
Funding Information
This work was partially supported by National Science Foundation Graduate Research Fellowship DGE-1247392 (2017-2021) and National Science Foundation KY-WV LSAMP Bridge to Doctorate Fellowship HRD-2004710 (2020-2021).
Recommended Citation
Vindas Meléndez, Andrés R., "Combinatorial Invariants of Rational Polytopes" (2021). Theses and Dissertations--Mathematics. 82.
https://uknowledge.uky.edu/math_etds/82
