On the generation of some Lie-type geometries

Let $X_n(K)$ be a building of Coxeter type $X_n = A_n$ or $X_n = D_n$ defined over a given division ring $K$ (a field when $X_n = D_n$). For a non-connected set $J$ of nodes of the diagram $X_n$, let $Γ(K) = Gr_J(X_n(K))$ be the $J$-Grassmannian of $X_n(K)$. We prove that $Γ(K)$ cannot be generated over any proper sub-division ring $K_0$ of $K$. As a consequence, the generating rank of $Γ(K)$ is infinite when $K$ is not finitely generated. In particular, if $K$ is the algebraic closure of a finite field of prime order then the generating rank of $Gr_{1,n}(A_n(K))$ is infinite, although its embedding rank is either $(n+1)^2-1$ or $(n+1)^2$.

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