  We study a number of conditions on the Hilbert function of a level artinian
algebra which imply the Weak Lefschetz Property (WLP). Possibly the most
important open case is whether a codimension 3 SI-sequence forces the WLP for
level algebras. In other words, does every codimension 3 Gorenstein algebra
have the WLP? We give some new partial answers to this old question: we prove
an affirmative answer when the initial degree is 2, or when the Hilbert
function is relatively small. Then we give a complete answer to the question of
what is the largest socle degree forcing the WLP.
