A Note on a Conjecture Regarding the Weak Lefschetz Property of a Special Class of Artinian Algebras

Let R=K[x1,…,xr]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R = K[x_1,\ldots ,x_r]$$\end{document} be the polynomial ring over an infinite field K. For a class of Artinian K-algebras A=R/I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A = R/I$$\end{document}, where I is a monomial ideal of certain specific form and K has some positive characteristics, we examine the weak Lefschetz property of A for various choices of I. In particular, these results support parts of a conjecture by Migliore, Miró-Roig and Nagel in some positive characteristics, and reveal that another part of their conjecture is characteristic-dependent.