Let $V$ be an $n$-dimensional vector space over a field $F$. Let $\mathrm{Gr}(V,d)$ be the set (Grassmann variety) of all $d$-dimensional subspaces of $V$. We can regard $\mathrm{Gr}(V,d)$ as a subset of the vector space $\Lambda^d(V)$ (of dimension ${n\choose d}$) via the Plücker embedding. Thus $\mathrm{Gr}(V,d)$ defines a matroid $\mathrm{Pl}(V,d)$ (where independence corresponds to linear independence in $\Lambda^d(V)$). Can anything interesting be said about this matroid?

In particular, there is a "natural" matroid structure on $\mathrm{Gr}(V,d)$, namely, the matroid corresponding to the $d$th Dilworth truncation $D_d \mathbb{B}(V)$ of the lattice $\mathbb{B}(V)$ of all subspaces of $V$. Is $D_d \mathbb{B}(V)$ isomorphic to $\mathrm{Pl}(V,d)$?