Geometric Properties of Weighted Projective Space Simplices

A long-standing conjecture in geometric combinatorics entails the interplay between three properties of lattice polytopes: reflexivity, the integer decomposition property (IDP), and the unimodality of Ehrhart h*-vectors. Motivated by this conjecture, this dissertation explores geometric properties of weighted projective space simplices, a class of lattice simplices related to weighted projective spaces. In the first part of this dissertation, we are interested in which IDP reflexive lattice polytopes admit regular unimodular triangulations. Let Delta(1,q)denote the simplex corresponding to the associated weighted projective space whose weights are given by the vector (1,q). Focusing on the case where Delta(1,q) is IDP reflexive such that q has two distinct parts, we establish a characterization of the lattice points contained in Delta(1,q) and verify the existence of a regular unimodular triangulation of its lattice points by constructing a Grobner basis with particular properties. In the second part of this dissertation, we explore how a necessary condition for IDP that relaxes the IDP characterization of Braun, Davis, and Solus yields a natural parameterization of an infinite class of reflexive simplices associated to weighted projective spaces. This parametrization allows us to check the IDP condition for reflexive simplices having high dimension and large volume, and to investigate h* unimodality for the resulting IDP reflexives in the case that Delta(1,q) is 3-supported

Projective Normality and Ehrhart Unimodality for Weighted Projective Space Simplices

Within the intersection of Ehrhart theory, commutative algebra, and algebraic geometry lie lattice polytopes. Ehrhart theory is concerned with lattice point enumeration in dilates of polytopes; lattice polytopes provide a sandbox in which to test many conjectures in commutative algebra; and many properties of projectively normal toric varieties in algebraic geometry are encoded through corresponding lattice polytopes. In this article we focus on reflexive simplices and work to identify when these have the integer decomposition property (IDP), or equivalently, when certain weighted projective spaces are projectively normal. We characterize the reflexive, IDP simplices whose associated weighted projective spaces have one projective coordinate with weight fixed to unity and for which the remaining coordinates can assume one of three distinct weights. We show that several subfamilies of such reflexive simplices have unimodal $h^\ast$-polynomials, thereby making progress towards conjectures and questions of Stanley, Hibi-Ohsugi, and others regarding the unimodality of their $h^\ast$-polynomials. We also provide computational results and introduce the notion of reflexive stabilizations to explore the (non-)ubiquity of reflexive simplices that are simultaneously IDP and $h^\ast$-unimodal

Triangulations, order polytopes, and generalized snake posets

This work regards the order polytopes arising from the class of generalized snake posets and their posets of meet-irreducible elements. Among generalized snake posets of the same rank, we characterize those whose order polytopes have minimal and maximal volume. We give a combinatorial characterization of the circuits in these order polytopes and then conclude that every regular triangulation is unimodular. For a generalized snake word, we count the number of flips for the canonical triangulation of these order polytopes. We determine that the flip graph of the order polytope of the poset whose lattice of filters comes from a ladder is the Cayley graph of a symmetric group. Lastly, we introduce an operation on triangulations called twists and prove that twists preserve regular triangulations.Comment: 39 pages, 26 figures, comments welcomed

Triangulations, Order Polytopes, and Generalized Snake Posets

This work regards the order polytopes arising from the class of generalized snake posets and their posets of meet-irreducible elements. Among generalized snake posets of the same rank, we characterize those whose order polytopes have minimal and maximal volume. We give a combinatorial characterization of the circuits in related order polytopes and then conclude that all of their triangulations are unimodular. For a generalized snake word, we count the number of flips for the canonical triangulation of these order polytopes. We determine that the flip graph of the order polytope of the poset whose lattice of upper order ideals comes from a ladder is the Cayley graph of a symmetric group. Lastly, we introduce an operation on triangulations called twists and prove that twists preserve regular triangulations.Mathematics Subject Classifications: 52B20, 52B05, 52B12, 06A07Keywords: Order polytopes, triangulations, flow polytopes, circuit

Ehrhart Theory of Paving and Panhandle Matroids

We show that the base polytope $P_M$ of any paving matroid $M$ can be obtained from a hypersimplex by slicing off subpolytopes. The pieces removed are base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams, whose Ehrhart polynomials we can calculate explicitly. Consequently, we can write down the Ehrhart polynomial of $P_M$, starting with Katzman's formula for the Ehrhart polynomial of a hypersimplex. The method builds on and generalizes Ferroni's work on sparse paving matroids. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of stressed-hyperplane relaxation introduced by Ferroni, Nasr, and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. We present evidence that panhandle matroids are Ehrhart positive and describe a conjectured combinatorial formula involving chain gangs and Eulerian numbers from which Ehrhart positivity of panhandle matroids will follow. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show that they depend only on their design-theoretic parameters: for example, while projective planes of the same order need not have isomorphic matroids, their base polytopes must be Ehrhart equivalent