Determinantal Ideals and Lefschetz Properties

The algebraic Lefschetz properties, inspired by the Hard Lefschetz Theorem in the cohomology of projective varieties, provide a natural framework for investigating the structure of graded algebras. Motivated by Murai’s question concerning the preservation of Lefschetz properties under squarefree Groebner degenerations, we study the weak and strong Lefschetz properties for the Stanley–Reisner ring R/in(I_t), where I_t is the ideal in a polynomial ring R generated by the t-minors of an mxn matrix of indeterminates, and in(I_t) denotes the initial ideal of It with respect to a diagonal monomial order. We show that when I_t is generated by maximal minors, that is, when t=min{m,n}, the algebra R/in(I_t) has the strong Lefschetz property for all m,n. In contrast, for t<min{m,n}, we establish a bound such that R/in(I_t) fails to satisfy the weak Lefschetz property whenever the product mn exceeds this bound. These results provide a negative answer to Murai’s question. Furthermore, the investigation of these Lefschetz properties led us to explore other algebraic questions and obtain additional results. In this talk, we will present our results on the weak and strong Lefschetz properties, show their implications for Murai’s question, and discuss some of the additional algebraic results that emerged along the way.