in certain cases deciding the presence of the weak Lefschetz property is equivalent to determining the splitting type of some semistable rank three vector bundles on the projective plane.

We now describe the contents of this paper in more detail. Let $R = K[x_1, \ldots, x_n]$ be the standard graded $n$-variate polynomial ring over the infinite field $K$, and let $A$ be a standard graded $K$-algebra over $R$. We say $A$ is [*artinian*]{} if $A$ is finite dimensional as a vector space over $K$. Further, an artinian $K$-algebra $A$ is said to have the [*weak Lefschetz property*]{} if there exists a linear form $\ell \in [A]_1$ such that, for all integers $d$, the multiplication map $\times \ell: [A]_d \rightarrow [A]_{d+1}$ has maximal rank, that is, the map is injective or surjective. Such a linear form is called a [*Lefschetz element*]{} of $A$.

The weak Lefschetz property has been studied extensively for many reasons, especially for the relation to the Hilbert function (see, e.g., [@BMMNZ], [@HMNW], [@MZ], and [@ZZ]). A convenient way to encode the Hilbert function of an artinian $K$-algebra $A$ is the [*$h$-vector*]{}, a finite sequence $h(A) = (h_0, \ldots, h_e)$ of positive integers $h_i = \dim_K{[A]_i}$. Using this 