On the rank varieties of some simple modules for symmetric groups

In the previous work, Lim and the author determined the rank variety of the simple $\mathbb{F}\mathfrak{S}_{kp}$-module $D(p-1)=D^{(kp-p+1,1^{p-1})}$ with respect to some maximal elementary abelian $p$-subgroup $E_k$ and the complexity when $k\not\equiv 1\pmod p$ and $p$ is odd. Their method relied on the dimension of the module, which is dependent on $k$. In this paper, we extend this result to the case for any $k\geq 2$, and determine the rank variety of $D(p-1)$ and its complexity, providing a proof independent of $k$.