The van der Waerden Simplicial Complex and the Lefschetz Properties

Abstract

In this thesis, we examine the van der Waerden simplicial complexes, defined on two parameters n and k, and the Artinian rings we can form by modding out the squares of the variables from the Stanley-Reisner ring of the van der Waerden complex. We begin by providing a thorough background on all the necessary background, from abstract algebra, graph theory, and linear algebra. The main question investigated in this thesis is when the Artinian rings constructed from the van der Waerden complexes have either the Weak or Strong Lefschetz property. We examine for what values of n and k do these Artinian rings have the Weak, and in some instances Strong, Lefschetz Property. We focus on the smallest possible values of k, namely k = 1 and k = 2; and the largest possible value k can take, which is n − 1. We also focus on the case where k = 3, as the first instance of the failure of the Weak Lefschetz Property occurs here.

We also investigate in what degrees the Artinian ring always has the Weak Lefschetz Property. We then give a characterization of when these simplicial complexes are pseudo- manifolds, which provides some further insight on what degrees the associated Artinian ring has the Weak Lefschetz Property. We conclude by providing some conjectures on both Lefschetz Properties, as well as further areas of possible future research. A Macaulay2 package on the van der Waerden simplicial complexes is also provided.