On the Weak Lefschetz Property for certain ideals generated by powers of linear forms

Ideals $I\subseteq R=k[\mathbb P^n]$ generated by powers of linear forms arise, via Macaulay duality, from sets of fat points $X\subseteq \mathbb P^n$. Properties of $R/I$ are connected to the geometry of the corresponding fat points. When the linear forms are general, many authors have studied the question of whether or not $R/I$ has the Weak Lefschetz Property (WLP). We study this question instead for ideals coming from a family of sets of points called grids. We give a complete answer in the case of uniform powers of linear forms coming from square grids, and we give a conjecture and approach for the case of nonsquare grids. In the cases where WLP holds, we also describe the non-Lefschetz locus.