On the Lefschetz Property for quotients by monomial ideals containing squares of variables

Let $\mathrm{\Delta }$ be an (abstract) simplicial complex on $n$ vertices. One can define the Artinian monomial algebra $A(\Delta) = \Bbbk[x_1, \ldots, x_n]/ \langle x_1^2, \ldots, x_n^2, I_{\Delta} \rangle$,where$\Bbbk$ is a field of characteristic $0$ and $I_{\mathrm{\Delta }}$ is the Stanley-Reisner ideal associated to $\mathrm{\Delta }$. In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of $A(\mathrm{\Delta })$ in terms of the simplicial complex $\mathrm{\Delta }$. We are able to completely analyze when WLP holds in degree $1$, complementing work by Migliore, Nagel and Schenck in [MNS2020]. We give a complete characterization of all $2$-dimensional pseudomanifolds $\mathrm{\Delta }$ such that $A(\mathrm{\Delta })$ satisfies WLP. We also construct Artinian Gorenstein algebras that fail WLP by combining our results and the standard technique of Nagata idealization.