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 ? be the building of type B n formed by the subspaces of V totally singular for q and, for 1?k?n, let ? k be the k-grassmannian of ?. Let ? k be the embedding of ? k into PG(? k V) mapping every point ?v 1,v 2,,v k ? of ? 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 2 or a number field, n>k and k=2 or 3, then ? k is universal.
orthogonal grassmannians weyl modules veronesean embeddings orthogonal groups