Grassmann and Weyl embeddings of orthogonal grassmannians

Abstract

Given a non-singular quadratic form q of maximal Witt index on V := V (2n + 1, F), let Δ be the building of type B n formed by the subspaces of V totally singular for q and, for 1k . In this paper we give a new very easy proof of this fact. We also prove that if char(F) = 2 then dim(As a consequence, when 1 < k < n and char(F) = 2 the embedding ε 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 ε k is universal.

Grassmann embeddings of polar Grassmannians

“…Furthermore, still assuming d 0 = 1, if F is a perfect field of characteristic 2 then Q n ∼ = S n = S n (n, 0; F). In this case the Grassmann embedding of S n yields a projective embedding of Q n which, as proved in [4], is a quotient of ε n . A 2 n -dimensional projective embedding also exists for Q n when d 0 = 2 (see e.g.…”

“…If d 0 = 1 then Q n admits the so-called spin embedding, which is projective and 2 n -dimensional. Interesting relations exist between this embedding and ε n (see [4], [5]; also Section 7.3 of this paper). Furthermore, still assuming d 0 = 1, if F is a perfect field of characteristic 2 then Q n ∼ = S n = S n (n, 0; F).…”

“…The dimension of ε k is known only for d 0 ≤ 1 with the further restriction k < n when d 0 = 0. Indeed, in [4] it is shown that…”

“…Remark 1.4. As noticed in Subsection 1.1, the dual polar space Q n (n, 0, 0; F) is not considered in [4]. Part 3 of the above theorem includes this case too.…”

Some results on caps and codes related to orthogonal Grassmannians — a preview

“…Properties of Grassmann and Veronese-spin embedding, fundamental in order to obtain our results, are extensively investigated in [5] and [6].…”