  An SI-sequence is a finite sequence of positive integers which is symmetric,
unimodal and satisfies a certain growth condition. These are known to
correspond precisely to the possible Hilbert functions of Artinian Gorenstein
algebras with the Weak Lefschetz Property, a property shared by most Artinian
Gorenstein algebras. Starting with an arbitrary SI-sequence, we construct a
reduced, arithmetically Gorenstein configuration $G$ of linear varieties of
arbitrary dimension whose Artinian reduction has the given SI-sequence as
Hilbert function and has the Weak Lefschetz Property. Furthermore, we show that
$G$ has maximal graded Betti numbers among all arithmetically Gorenstein
subschemes of projective space whose Artinian reduction has the Weak Lefschetz
Property and the given Hilbert function. As an application we show that over a
field of characteristic zero every set of simplicial polytopes with fixed
$h$-vector contains a polytope with maximal graded Betti numbers.
