Grassmann and Weyl embeddings of orthogonal grassmannians

Given a non-singular quadratic form $q$ of maximal Witt index on $V := V(2n+1,\F)$, let $\Delta$ be the building of type $B_n$ formed by the subspaces of $V$ totally singular for $q$ and, for $1\leq k \leq n$, let $\Delta_k$ be the $k$-grassmannian of $\Delta$. Let $\varepsilon_k$ be the embedding of $\Delta_k$ into $\PG(\bigwedge^kV)$ mapping every point $\langle v_1,v_2,...,v_k\rangle$ of $\Delta_k$ to the point $\langle v_1\wedge v_2\wedge ...\wedge v_k\rangle$ of $\PG(\bigwedge^k V)$. It is known that if $\mathrm{char}(\F)\neq 2$ then $\mathrm{dim}(\varepsilon_k)={{2n+1}\choose k}$. In this paper we give a new very easy proof of this fact. We also prove that if $\mathrm{char}(\F) = 2$ then $\mathrm{dim}(\varepsilon_k)={{2n+1}\choose k}-{{2n+1}\choose {k-2}}$. As a consequence, when $1 < k < n$ and $\mathrm{char}(\F) = 2$ the embedding $\varepsilon_k$ is not universal. Finally, we prove that if $\F$ is a perfect field of characteristic $p > 2$ or a number field, $n > k$ and $k = 2$ or $3$, then $\varepsilon_k$ is universal.