  Deciding the presence of the weak Lefschetz property often is a challenging
problem. In this work an in-depth study is carried out in the case of Artinian
monomial ideals with four generators in three variables. We use a connection to
lozenge tilings to describe semistability of the syzygy bundle of such an
ideal, to determine its generic splitting type, and to decide the presence of
the weak Lefschetz property. We provide results in both characteristic zero and
positive characteristic.
