A tour of the weak and strong Lefschetz properties (Q393095)

In 1980, \textit{R. P. Stanley} [SIAM J. Algebraic Discrete Methods 1, 168--184 (1980; Zbl 0502.05004)] proved the following theorem: Let \(R = k[x_1, \dots, x_r]\), where \(k\) has characteristic zero. Let \(I = (x_1^{a_1}, \dots, x_r^{a_r})\). Let \(\ell\) be a general linear form. Then, for any positive integers \(d\) and \(i\), the homomorphism induced by multiplication by \(\ell^d\), \(\times \ell^d: [R/I]_i \rightarrow [R/I]_{i+d}\), has maximal rank. This theorem has been reproved by a number of mathematicians using a variety of techniques and has motivated the study of the weak and strong Lefschetz properties. Let \(A = R/I\) be a graded artinian algebra, where \(k\) is an infinite field, and let \(\ell\) be a general linear form. \(A\) is said to have the \textit{weak Lefschetz property} (WLP) if the homomorphism induced by multiplication by \(\ell\), \(\times \ell: A_i \rightarrow A_{i+1}\), has maximal rank for all \(i\) (i.e., is injective or surjective). Further, we say that \(A\) has the \textit{strong Lefschetz property} if \(\times \ell^d: A_i \rightarrow A_{i+d}\), has maximal rank for all \(i\) and \(d\) (i.e., is injective or surjective).NEWLINENEWLINEThis paper is a survey of the different directions and research that has resulted from studying these properties. The directions highlight the intersection of commutative algebra, algebraic geometry, and combinatorics. After providing some background, the paper is divided into sections of cases for the study: complete intersections and Gorenstein algebras; monomial level algebras, powers of linear forms, connections between Fröberg's conjecture and the WLP, and positive characteristics and enumerations (involving determinants of certain matrices, lozenge tilings of punctured hexagons, and perfect matchings of bipartite graphs). Throughout the paper, the authors include a number of open questions that drive current research programs of many experts.