The van der Waerden Simplicial Complex and its Lefschetz Properties

Abstract

The van der Waerden simplicial complex, denoted $\mathtt{vdw}(n,k)$, is the simpicial complex whose facets correspond to the arithmetic progressions of length $k$ in the set $\{1,\dots ,n\}$. We study the Lefschetz properties of the Artinian ring $A(n,k)=K[x_1,\dots ,x_n]/(I_{\mathtt{vdw}(n,k)}+⟨x_1^2,\dots ,x_n^2⟩)$ where $I_{\mathtt{vdw}(n,k)}$ 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 $n\gg k\ge 3$ and $k$ odd. We also classify when $\mathtt{vdw}(n,k)$ 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.

21 pages