Grassmann and Weyl embeddings of orthogonal grassmannians

Given a non-singular quadratic form q of maximal Witt index on $V := V(2n+1,\mathbb{F})$ , let Δ be the building of type B _n formed by the subspaces of V totally singular for q and, for 1≤k≤n, let Δ _k be the k-grassmannian of Δ. Let ε _k be the embedding of Δ _k into PG(? ^k V) mapping every point 〈v ₁,v ₂,…,v _k 〉 of Δ _k to the point 〈v ₁∧v ₂∧?∧v _k 〉 of PG(? ^k V). It is known that if $\mathrm{char}(\mathbb{F})\neq2$ 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 $\mathrm{char}(\mathbb{F}) = 2$ then $\mathrm{dim}(\varepsilon_{k})={{2n+1}\choose k}-{{2n+1}\choose{k-2}}$ . As a consequence, when 1<k<n and $\mathrm{char}(\mathbb{F}) = 2$ the embedding ε _k is not universal. Finally, we prove that if $\mathbb{F}$ is a perfect field of characteristic p>2 or a number field, n>k and k=2 or 3, then ε _k is universal.