The strong Lefschetz property of certain modules over Clements-Lindström rings

We introduce a method for studying the Lefschetz properties for k [ x , y ] k[x,y] -modules based on the Lindström-Gessel-Viennot Lemma. We prove that certain modules over Artinian Clements-Lindström rings in characteristic zero have the strong Lefschetz property. In particular, we show that every homogeneous idea in a Clements-Lindström ring of embedding dimension two has the strong Lefschetz property. As an application, we study the strong Lefschetz property of type two monomial ideals of codimension three.