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University of Kentucky UKnowledge Theses and Dissertations--Mathematics Mathematics 2021 Combinatorial Invariants of Rational Polytopes Andrés R. Vindas Meléndez University of Kentucky, [email protected] Author ORCID Identifier: https://orcid.org/0000-0002-7437-3745 Digital Object Identifier: https://doi.org/10.13023/etd.2021.225 Right click to open a feedback form in a new tab to let us know how this document benefits ou.y Recommended Citation Vindas Meléndez, Andrés R., "Combinatorial Invariants of Rational Polytopes" (2021). Theses and Dissertations--Mathematics. 82. https://uknowledge.uky.edu/math_etds/82 This Doctoral Dissertation is brought to you for free and open access by the Mathematics at UKnowledge. It has been accepted for inclusion in Theses and Dissertations--Mathematics by an authorized administrator of UKnowledge. For more information, please contact [email protected]. STUDENT AGREEMENT: I represent that my thesis or dissertation and abstract are my original work. Proper attribution has been given to all outside sources. I understand that I am solely responsible for obtaining any needed copyright permissions. I have obtained needed written permission statement(s) from the owner(s) of each third-party copyrighted matter to be included in my work, allowing electronic distribution (if such use is not permitted by the fair use doctrine) which will be submitted to UKnowledge as Additional File. I hereby grant to The University of Kentucky and its agents the irrevocable, non-exclusive, and royalty-free license to archive and make accessible my work in whole or in part in all forms of media, now or hereafter known. I agree that the document mentioned above may be made available immediately for worldwide access unless an embargo applies. I retain all other ownership rights to the copyright of my work. I also retain the right to use in future works (such as articles or books) all or part of my work. I understand that I am free to register the copyright to my work. REVIEW, APPROVAL AND ACCEPTANCE The document mentioned above has been reviewed and accepted by the student’s advisor, on behalf of the advisory committee, and by the Director of Graduate Studies (DGS), on behalf of the program; we verify that this is the final, approved version of the student’s thesis including all changes required by the advisory committee. The undersigned agree to abide by the statements above. Andrés R. Vindas Meléndez, Student Dr. Benjamin Braun, Major Professor Dr. Benjamin Braun, Director of Graduate Studies Combinatorial Invariants of Rational Polytopes DISSERTATION A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the College of Arts and Sciences at the University of Kentucky By Andr´esR. Vindas Mel´endez Lexington, Kentucky Director: Dr. Benjamin Braun, Professor of Mathematics Lexington, Kentucky 2021 Copyright c Andr´esR. Vindas Mel´endez2021 https://orcid.org/0000-0002-7437-3745 ABSTRACT OF DISSERTATION Combinatorial Invariants of Rational Polytopes 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 ra- tional 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 permutahe- dron. 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 quasipoly- nomial 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 decom- positions 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. KEYWORDS: rational polytope, Ehrhart quasipolynomial, lattice points Andr´esR. Vindas Mel´endez May 24, 2021 Combinatorial Invariants of Rational Polytopes By Andr´esR. Vindas Mel´endez Dr. Benjamin Braun Director of Dissertation Dr. Benjamin Braun Director of Graduate Studies May 24, 2021 Para mis creadores, Mami (Sara) y Papi (Rodolfo). Para mis hermanas, Alejandra y Sarah. Para mis padrinos, Ana y Julio. Para mi familia con la que el universo me bendici´o(mis t´ıos,t´ıas,primos, y sobrinos) y para mis amigos que se han convertido en familia. Para mis maestros que me ense~narona apreciar la belleza de la matem´atica. For my creators, Mami (Sara) and Papi (Rodolfo). For my sisters, Alejandra and Sarah. For my godparents, Ana and Julio. For my family with whom the universe blessed me with (my uncles, aunts, cousins, and nephews) and for my friends who have become family. For my teachers who taught me to appreciate the beauty of math. ACKNOWLEDGMENTS Gracias a la vida que me ha dado tanto. Thank you to everyone who has crossed paths with me and has helped me redefine what it means to be a mathematician. Thank you to my influential teachers and mentors who have dedicated their time to my success. Thank you to all my friends that I have made at the University of Kentucky and in the mathematics department. Friendship is something I value, so thank you to all my friends that I have created throughout my life. To my friends who have become family (you know who you are), thank you for your love and support. A very special thank you to my doctoral advisor, Benjamin Braun. Thank you for listening to my passions, ideas, and concerns. Thank you for being generous with your time, for growing alongside me, and for acknowledging that I have something to offer mathematics. The kindness you showed me even before you met me in person confirmed that I would flourish with your support. I also extend a thank you to my master's thesis advisors, Federico Ardila and Matthias Beck, who agreed to continue to serve as research mentors as part of the California Doctoral Incentive Program. Their support, friendship, and continued collaboration is invaluable. Thank you also to Cindy Jong, Uwe Nagel, Martha Yip, and Derek Young for serving on my committee. iii Thank you to everyone who is part of my math community. I do not envision myself doing math if I cannot do it in community and with friends. Thank you to the Creator and my creators, Mami and Papi. I also thank my sisters Alejandra and Sarah, my godparents t´ıaAna and t´ıoJulio, my cousins, and the rest of my family: gracias por su amor y por apoyar mis metas. Este logro es de todos. <Pura vida! iv TABLE OF CONTENTS Acknowledgments . iii List of Tables . vi List of Figures . vii Chapter 1 Introduction . 1 Chapter 2 Background . 6 2.1 Lattice and Rational Polytopes . 6 2.2 Ehrhart theory . 12 Chapter 3 The Equivariant Ehrhart Theory of the Permutahedron . 22 3.1 Equivariant Volumes of the Permutahedron . 22 3.2 Equivariant Ehrhart Theory of the Permutahedron . 30 Chapter 4 Decompositions of the Ehrhart h∗-polynomial for Rational Poly- topes . 45 4.1 Rational Betke{McMullen Decomposition . 48 4.2 h∗-Decompositions from Boundary Triangulations . 51 4.3 Applications . 58 Bibliography . 63 Vita . 69 v LIST OF TABLES 3.1 λ-compatibility for m =3........................... 36 ∗ 3.2 The equivariant H -series of Π3 ....................... 39 ∗ 3.3 The equivariant H -series of Π4 ....................... 39 4.1 Table for the Ω 2 T; dim(Ω);B(Ω; z), and h(Ω; zL). 62 vi LIST OF FIGURES 1.1 The platonic solids (from left to right): tetrahedron, cube, octahedron, dodecahedron, and icosahedron. 1 1.2 The triangle T ................................. 2 1.3 The triangle T with some of its integer dilates. 3 2.1 A hexagon as a bounded intersection of 6 half-spaces (left) and a hexagon as the convex hull of 6 vertices (right). 7 2.2 (Left to right): 1-simplex (line segment), 2-simplex (triangle), 3-simplex (tetrahedron). 8 2.3 A triangulation of the 3-cube into four 3-simplices. 8 2.4 The Minkowski sum of [(0; 0); (1; 0)], [(0; 0); (0; 1)], and [(0; 0); (1; 1)]. 9 2.5 Z((1; 0; 0); (0; 1; 0); (0; 0; 1); (1; 1; 1)). 9 2.6 A zonotopal decomposition of Z((0; 1); (1; 0); (1; 1). 10 2.7 The permutahedron Π4. ........................... 11 2 2.8 The square ⊂ Z ............................... 12 2.9 The square and its second dilate 2. .................. 14 2.10 The unit square P = [0; 1]2 ⊂ Z2 and some of its dilates. 15 1 2 2 2.11 The rational rectangle P = [0; 2 ] ⊂ Z and some of its dilates. 16 2.12 The zonotope Z((0; 1); (1; 0); (1; 1); (1; −1)) is a Minkowski sum of four line segments. 20 2.13 A zonotopal decomposition of Π3 with corresponding labeled forests. 21 (12) 3.1 The slice Π4 of the permutahedron Π4 fixed by (12) 2 S4 is a hexagon. 23 (12) 3.2 The fixed polytope Π4 is a half-integral hexagon containing 6 lattice points. 31 (12) 3.3 Decomposition of the fixed polytope Π4 into half-open parallelepipeds.
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  • Triangulations of Integral Polytopes, Examples and Problems

    Triangulations of Integral Polytopes, Examples and Problems

    数理解析研究所講究録 955 巻 1996 年 133-144 133 Triangulations of integral polytopes, examples and problems Jean-Michel KANTOR We are interested in polytopes in real space of arbitrary dimension, having vertices with integral co- ordinates: integral polytopes. The recent increase of interest for the study of these $\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}$ topes and their triangulations has various motivations; let us mention the main ones: . the beautiful theory of toric varieties has built a bridge between algebraic geometry and the combina- torics of these integral polytopes [12]. Triangulations of cones and polytopes occur naturally for example in problems of existence of crepant resolution of singularities $[1,5]$ . The work of the school of $\mathrm{I}.\mathrm{M}$ . Gelfand on secondary polytopes gives a new insight on triangulations, with applications to algebraic geometry and group theory [13]. In statistical physics, random tilings lead to some interesting problems dealing with triangulations of order polytopes $[6,29]$ . With these motivations in mind, we introduce new tools: Generalizations of the Ehrhart polynomial (counting points “modulo congruence”), discrete length between integral points (and studying the geometry associated to it), arithmetic Euler-Poincar\'e formula which gives, in dimension 3, the Ehrhart polynomial in terms of the $\mathrm{f}$-vector of a minimal triangulation of the polytope (Theorem 7). Finally, let us mention the results in dimension 2 of the late Peter Greenberg, they led us to the study of “Arithmetical $\mathrm{P}\mathrm{L}$-topology” which, we believe with M. Gromov, D. Sullivan, and P. Vogel, has not yet revealed all its beauties.