  Let $\Delta$ 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 $I_\Delta$ is the Stanley-Reisner ideal associated to
$\Delta$. In this paper, we aim to characterize the Weak Lefschetz Property
(WLP) of $A(\Delta)$ in terms of the simplicial complex $\Delta$. 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 $\Delta$ such that $A(\Delta)$ satisfies
WLP. We also construct Artinian Gorenstein algebras that fail WLP by combining
our results and the standard technique of Nagata idealization.
