Lattice Polytopes - Applications and Properties: Ehrhart Theory, Graph Colorings, and Level Algebras

Abstract

This dissertation is about applications and properties of lattice polytopes. In the second chapter, we briefly review the necessary background material.

In Chapter 3, we examine proper colorings of a special family of graphs. This family is described by the Cartesian graph product of an arbitrary graph and a path (cycle) graph. It is important to note that there are two independent parameters, the size of the path (cycle) graph and the number of colors. In order to determine the number of proper colorings, we combine transfer-matrix methods with Ehrhart theory. Furthermore, we use group actions to limit the size of the transfer matrix. We describe an explicit formula for the chromatic polynomial of the product with a path graph. Moreover, we describe the asymptotic behavior in the case of the product with a cycle graph.

In Chapter 4, we examine and fully characterize the level property of order polytopes, where levelness is a generalization of the Gorenstein property. We use weighted, directed graphs to describe this characertization. This enables us to give construct an infinite class of level order polytopes. In addition, we characterize the level property of alcoved polytopes.

In Chapter 5, both the Gorenstein and the level property of s-lecture hall polytopes are described. We use the results from the literature about the Gorenstein property of s-lecture hall cones to describe and characterize the Gorenstein property of s-lecture hall polytopes. Furthermore, we give a characterization of the level property in terms of inversion sequences. This characterization can explicitly be applied in lower dimensions, which is illustrated while we prove that every 2-dimensional s-lecture hall polygon is level.

In Chapter 6, we examine the holes of affine semigroups. Based on a publication by Hemmecke, Takemura, and Yoshida, we develop a software that describes the set of holes. We use this software to examine the holes of semigroups coming from the common diagonal effect model. Based on explicit computations we conjecture and then prove the structure of these holes. Furthermore, we show that the 7th linear ordering polytope has the integer-decomposition property.

Chapter 7 can be seen as a manual of this software. The source code can be found in the appendix.