  Migliore-Mir\'o-Roig-Nagel [Trans. A.M.S. 2011, arXiv: 0811.1023] show that
the weak Lefschetz property (WLP) can fail for an ideal I in K[x_1,x_2,x_3,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 [H.Schenck, A.Seceleanu,
Proc. A.M.S. 2010, arXiv:0911.0876]. 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 [J. Eur. Math. Soc. 2010, arXiv:0803.0892]
allow us to relate WLP to Gelfand-Tsetlin patterns. See the paper "On the weak
Lefschetz property for powers of linear forms" by Migliore-Mir\'o-Roig-Nagel
[arXiv:1008.2149] for related results.
