The van der Waerden Simplicial Complex and its Lefschetz Properties

TL;DR

This paper investigates the Lefschetz properties of a simplicial complex related to arithmetic progressions, classifies cases with the Weak Lefschetz Property, and conjectures about its failure in certain parameters.

Contribution

It introduces the van der Waerden simplicial complex, studies its associated Artinian rings, classifies when they have the Weak Lefschetz Property, and proposes a conjecture for broader failure cases.

Findings

For k=1,2, or n-1, the ring has the Weak Lefschetz Property for all n > k.

Classified when A(n,3) has the Weak Lefschetz Property.

Identified conditions under which the complex is a pseudo-manifold, aiding Lefschetz property analysis.

Abstract

The van der Waerden simplicial complex, denoted ${\tt vdw}(n,k)$, is the simpicial complex whose facets correspond to the arithmetic progressions of length $k$ in the set $\{1,\ldots,n\}$. We study the Lefschetz properties of the Artinian ring $A(n,k) = K[x_1,\ldots,x_n]/(I_{{\tt vdw}(n,k)} + \langle x_1^2,\ldots,x_n^2\rangle)$ where $I_{{\tt vdw}(n,k)}$ is the associated Stanley--Reisner ideal. If $k=1,2$ or $n-1$, the ring $A(n,k)$ will have the Weak Lefschetz Property for all $n > k$. When $k=3$, we classify the rings $A(n,3)$ that have the Weak Lefschetz Property. We conjecture that $A(n,k)$ fails to have the Weak Lefschetz Property if $n \gg k \geq 3$ and $k$ odd. We also classify when ${\tt vdw}(n,k)$ is a pseudo-manifold, which allows us to show that $A(n,k)$ satisfies the Weak Lefschetz Property in some degrees by using a result of Dao and Nair.