A class of Hilbert series and the strong Lefschetz property

We determine the class of Hilbert series H H so that if M M is a finitely generated zero-dimensional R R -graded module with the strong Lefschetz property, then M ⊗ k k y / (y m) M ₖ ky/ (yᵐ) has the strong Lefschetz property for an indeterminate y y and all positive integers m m if and only if the Hilbert series of M M is in H H. Given two finite graded R R -modules M M and N N with the strong Lefschetz property, we determine sufficient conditions in order that M ⊗ k N M ₖN has the strong Lefschetz property.