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

Let Delta be an (abstract) simplicial complex on n vertices. One can define the Artinian monomial algebra A(Delta)=k[x1, horizontal ellipsis ,xn]/x12, horizontal ellipsis ,xn2,I Delta, where k 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 the WLP holds in degree 1, complementing work by Migliore, Nagel and Schenck on the WLP for quotients by quadratic monomials. We give a complete characterization of all 2-dimensional pseudomanifolds Delta such that A(Delta) satisfies the WLP. We also construct Artinian Gorenstein algebras that fail the WLP by combining our results and the standard technique of Nagata idealization.