The Hilbert functions which force the Weak Lefschetz Property

The purpose of this note is to characterize the finite Hilbert functions which force all of their artinian algebras to enjoy the Weak Lefschetz Property (WLP). Curiously, they turn out to be exactly those (characterized by Wiebe in [Wi]) whose Gotzmann ideals have the WLP. This implies that, if a Gotzmann ideal has the WLP, then all algebras with the same Hilbert function (and hence lower Betti numbers) have the WLP as well. However, we will answer in the negative, even in the case of level algebras, the most natural question that one might ask after reading the previous sentence: If A is an artinian algebra enjoying the WLP, do all artinian algebras with the same Hilbert function as A and Betti numbers lower than those of A have the WLP as well? Also, as a consequence of our result, we have another (simpler) proof of the fact that all codimension 2 algebras enjoy the WLP (this fact was first proven in [HMNW ], where it was shown that even the Strong Lefschetz Property holds). Let A = R/I be a standard graded artinian algebras, where R is a polynomial ring in r variables over a field k of characteristic zero, I is a homogeneous ideal of R, and the xi’s all have degree 1. We say that A enjoys the Weak Lefschetz Property (WLP) if, for a generic linear form L ∈ R and for all indices i ≥ 0, the multiplication map “·L” between the k-vector spaces Ai and Ai+1 has maximal rank (notice that, since A is artinian, Ai = 0