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

TL;DR

This paper investigates the failure of the weak Lefschetz property in certain polynomial ideals by connecting algebraic properties to geometric configurations of fat points and Gelfand-Tsetlin patterns.

Contribution

It introduces a novel approach using inverse systems to relate WLP failure to the geometry of fat point schemes and Gelfand-Tsetlin patterns.

Findings

WLP can fail for ideals generated by powers of linear forms in four variables.

Failure of WLP is linked to the geometry of associated fat point schemes.

Gelfand-Tsetlin patterns provide a combinatorial perspective on WLP conditions.

Abstract

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.