We give a criterion for an Artinian graded module over a polynomial ring in two variables to have weak Lefschetz property or not by using the classification of indecomposable representations of the 2-Kronecker quiver. As an application, we can recover the well-known result that all principal Artinian graded modules over a polynomial ring in two variables have weak Lefschetz properties.
https://ror.org/01c697313 
