Generic initial ideals of Artinian ideals having Lefschetz properties or the strong Stanley property

Abstract

For a standard Artinian k-algebra A = R / I , we give equivalent conditions for A to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of gin ( I ) . Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of gin ( I ) are determined just by the Hilbert function of I if A has the weak Lefschetz property. Furthermore, for the case that A is a standard Artinian k-algebra of codimension 3, we show that every graded Betti number of gin ( I ) is determined by the graded Betti numbers of I if A has the weak Lefschetz property. And if A has the strong Lefschetz (respectively Stanley) property, then we show that the minimal system of generators of gin ( I ) is determined by the graded Betti numbers (respectively by the Hilbert function) of I.