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

We study the weak and strong Lefschetz properties for $R\mathrm{/}\mathrm{i}\mathrm{n}(I_t)$, 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 $\mathrm{i}\mathrm{n}(I_t)$ 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, $t=\mathrm{m}\mathrm{i}\mathrm{n}\{m,n\}$), the ring $R\mathrm{/}\mathrm{i}\mathrm{n}(I_t)$ has the strong Lefschetz property for all $m$, $n$. In contrast, for $t<\mathrm{m}\mathrm{i}\mathrm{n}\{m,n\}$, we provide a bound such that $R\mathrm{/}\mathrm{i}\mathrm{n}(I_t)$ 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 Gröbner degenerations.