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

Abstract

In a recently published paper [Trans. Amer. Math. Soc. 363 (2011) 229–257], Migliore, Miró-Roig and Nagel show that the weak Lefschetz property (WLP) can fail for an ideal I ⊆ 𝕜[x1, …, x4] generated by powers of linear forms. This is in contrast to the analogous situation in 𝕜[x1, x2, x3], where WLP always holds [H. Schenck and A. Seceleanu, Proc. Amer. Math. Soc. 138 (2010) 2335–2339]. We use the inverse system dictionary to connect I to an ideal of fat points, and show that the failure of WLP for powers of linear forms is connected to the geometry of the associated fat point scheme. Recent results of Sturmfels and Xu [J. Eur. Math. Soc. 12 (2010) 429–459] allow us to relate the WLP to Gelfand–Tsetlin patterns.