5 research outputs found
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 -polynomials, thereby making progress towards
conjectures and questions of Stanley, Hibi-Ohsugi, and others regarding the
unimodality of their -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
-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
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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 of any paving matroid 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 , 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