The van der Waerden Simplicial Complex and its Lefschetz Properties

The van der Waerden simplicial complex, denoted $\mathtt{v}\mathtt{d}\mathtt{w}(n,k)$, 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 $A(\mathtt{v}\mathtt{d}\mathtt{w}(n,k))=K[x_1,\dots ,x_n]\mathrm{/}(I_{\mathtt{v}\mathtt{d}\mathtt{w}(n,k)}+⟨x_1^2,\dots ,x_n^2⟩)$ where $I_{\mathtt{v}\mathtt{d}\mathtt{w}(n,k)}$ is the associated Stanley–Reisner ideal. If $k=1,2$ or $n-1$, the ring $A(\mathtt{v}\mathtt{d}\mathtt{w}(n,k))$ will have the Weak Lefschetz Property for all $n>k$. When $k=3$, we classify the rings $A(\mathtt{v}\mathtt{d}\mathtt{w}(n,3))$ that have the Weak Lefschetz Property when the characteristic is zero. We conjecture that $A(\mathtt{v}\mathtt{d}\mathtt{w}(n,k))$ fails to have the Weak Lefschetz Property if $n\gg k\ge 3$ and $k$ odd. We also classify when $\mathtt{v}\mathtt{d}\mathtt{w}(n,k)$ is a pseudo-manifold, which allows us to show that $A(\mathtt{v}\mathtt{d}\mathtt{w}(n,k))$ satisfies the Weak Lefschetz Property in some degrees by using a result of Dao and Nair.