Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property

In [19], Migliore–Miró-Roig–Nagel show that the Weak Lefschetz property can fail for an ideal I ⊆ K[x 1 ,. .. , x 4 ] generated by powers of linear forms. This is in contrast to the analogous situation in K[x 1 , x 2 , x 3 ], where WLP always holds [24]. We use the inverse system dictionary to connect I to an ideal of fat points, and show that failure of WLP for powers of linear forms is connected to the geometry of the associated fat point scheme. Recent results of Sturmfels-Xu in [26] allow us to relate WLP to Gelfand-Tsetlin patterns.