Saturation rank for nilradical of parabolic subalgebras in Type A

Let $\mfp(d)$ be a standard parabolic subalgebra of $\mfsl_{n+1}(K)$ and $\mfu$ be the corresponding nilradical defined over an algebraically closed field $K$ of characteristic $p>0$. We construct a finite connected quiver $Q(d)$, through which we provide a combinatorial characterization of the centralizer $c_{\mfu}(x(d))$ of the Richardson element $x(d)$. We specifically focus on the centralizer when the levi factor of $\mfp(d)$ is determined by either one or two simple roots. This allows us to demonstrate that, under certain mild restrictions, the saturation rank of $\mfu$ equals the semisimple rank of the algebraic $K$-group $\SL_{n+1}(K)$.