On the weak and strong Lefschetz properties for initial ideals of determinantal ideals with respect to diagonal monomial orders

TL;DR

This paper investigates the Lefschetz properties of initial ideals of determinantal ideals, showing that maximal minors always satisfy the strong Lefschetz property, while smaller minors may fail, providing counterexamples to a related conjecture.

Contribution

It establishes conditions under which initial ideals of determinantal ideals have the Lefschetz properties and presents counterexamples to a conjecture about their preservation under degenerations.

Findings

Maximal minors' initial ideals have the strong Lefschetz property.

For smaller minors, Lefschetz properties may fail beyond certain bounds.

Counterexamples to Murai's question on Lefschetz property preservation.

Abstract

We study the weak and strong Lefschetz properties for $R/\mathrm{in}(I_t)$, where $I_t$ is the ideal of a polynomial ring $R$ generated by the $t$-minors of an $m\times n$ matrix of indeterminates, and $\mathrm{in}(I_t)$ denotes the initial ideal of $I_t$ with respect to a diagonal monomial order. We show that when $I_t$ is generated by maximal minors (that is, $t=\mathrm{min}\{m,n\}$), the ring $R/\mathrm{in}(I_t)$ has the strong Lefschetz property for all $m$, $n$. In contrast, for $t<\mathrm{min}\{m,n\}$, we provide a bound such that $R/\mathrm{in}(I_t)$ fails to satisfy the weak Lefschetz property whenever the product $mn$ exceeds this bound. As an application, we present counterexamples that provide a negative answer to a question posed by Murai regarding the preservation of Lefschetz properties under square-free Gr\"obner degenerations.