Let \( k \) be a field and \( A \) a (standard) graded Artinian \( k \)-algebra. \( A \) is said to have the Weak Lefschetz Property (WLP) if multiplication by a general linear form \( \times l \colon A_i \rightarrow A_{i+1} \) is full rank for all \( i \geq 0 \). We also say \( A \) has the Strong Lefschetz Property (SLP) if multiplication by powers of a general linear form \( \times l^d \colon A_i \rightarrow A_{i+d} \) is full rank for all \( i \geq 0 \) and for all \( d \geq 0\). Note that if \( A \) has the SLP, then \( A \) also has the WLP.

Although this is phrased algebraically, this turns out to be of use to combinatorialists as well. Of interest to many combinatorialists is if some sequence is unimodal (or log-concave). It turns out that the \( h \)-vector of an Artinian algebra with the WLP must be unimodal. So, a strategy for a combinatorialist to show that their sequence is unimodal is to construct an Artinian algebra and show that this algebra has the WLP.

Determining if an Artinian algebra \( A \) has the WLP seems like an innocent problem, but it turns out to be *extremely* hard to answer.
Even in the case of Artinian Gorenstein algebras, this is still a very difficult question.
Since this problem is phrased with a foundation in algebra, many of the methods to try to characterize such Artinian algebras rely on algebraic methods.

Exploiting the correspondence between algebra and geometry, some authors have tried to determine how the geometry of an object influences its corresponding Artinian algebra having the WLP. For instance,

While this page is under construction, please see Betti tables forcing failure of the Weak Lefschetz Property for some results.

Joint work with Hal Schenck.