Let $K$ be an algebraically closed field of characteristic
zero and let $I=(f_1 \komdots f_n)$ be a homogeneous
$R_+$-primary ideal in $R:=K[X,Y,Z]$. If the corresponding
syzygy bundle $\Syz(f_1 \komdots f_n)$ on the projective plane
is semistable, we show that the Artinian algebra $R/I$ has the
Weak Lefschetz property if and only if the syzygy bundle has a
special generic splitting type. As a corollary we get the
result of Harima et alt., that every Artinian complete
intersection ($n=3$) has the Weak Lefschetz
property. Furthermore, we show that an almost complete
intersection ($n=4$) does not necessarily have the Weak
Lefschetz property, answering negatively a question of
Migliore and Miró-Roig. We prove that an almost
complete intersection has the Weak Lefschetz property if the
corresponding syzygy bundle is not semistable.