There are many standard graded algebras over a field that are known or expected to have the weak Lefschetz property (WLP). There also examples of such algebras that do not have the WLP. This raises the question how common the WLP is. In the talk, focusing on quotients by monomial ideals, we discuss a framework for making this question precise and some initial results.

This is based on an ongoing project with Sonja Petrovic and Dane Wilburne.