The weak Lefschetz property, monomial ideals, and lozenges

By David D. Cook, Uwe Nagel. 2009

59 citations recorded by the source corpus.

We study the weak Lefschetz property and the Hilbert function of level Artinian monomial almost complete intersections in three variables. Several such families are shown to have the weak Lefschetz property if the characteristic of the base field is zero or greater than the maximal degree of any minimal generator of the ideal. Two of the families have an interesting relation to tilings of hexagons by lozenges. This lends further evidence to a conjecture by Migliore, Miro-Roig, and the second author. Finally, using our results about the weak Lefschetz property, we show that the Hilbert function of each level Artinian monomial almost complete intersection in three variables is peaked strictly unimodal.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/2747ec9512b51183a625e72aa68352d3db078c84

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Citations in the corpus

On the Weak Lefschetz Property for Powers of Linear Forms (2010, unknown): “wo authors [18]. In [19] we gave a much more extensive study of monomial almost complete intersections and when they fail to have the WLP. This work was extended by D. Cook II and the third author in [7]. In [4], we showed that the only other situation where level monomial ideals have to have the WL”
Survey Article: A tour of the weak and strong Lefschetz properties (2013, background): “Concerning the connection to the WLP, the following result of Migliore, Miró-Roig and Nagel gives a partial answer.”