Grassmann embeddings of polar Grassmannians

Abstract

In this paper we compute the dimension of the Grassmann embeddings of the polar Grassmannians associated to a possibly degenerate Hermitian, alternating or quadratic form with possibly non-maximal Witt index. Moreover, in the characteristic 2 case, when the form is quadratic and non-degenerate with bilinearization of minimal Witt index, we define a generalization of the so-called Weyl embedding (see [4]) and prove that the Grassmann embedding is a quotient of this generalized 'Weyl-like' embedding. We also estimate the dimension of the latter.

The generating rank of a polar Grassmannian

“…Let H be a non-degenerate Hermitian polar space of finite rank n and defect d and let H k be the k-Grassmannian of H, for 1 ≤ k ≤ n. Let F be the underlying field of H. By [7], for k < n the geometry H k affords a projective embedding ε k in PG( k V ) called the Plücker embedding, where V = V (N, F) is the vector space hosting the (unique) embedding of H. We have dim(ε k ) = dim( k V ) = N k . It follows from [21] that H k admits the universal embedding.…”

“…Indeed, since H k admits an embedding of dimension 2n+d k (namely the Plücker embedding [7]), we have er(H k ) ≥ 2n+d k . By combining this inequality with ( * ) we get the thesis of the theorem.…”

“…Proof. Put N := 2n + d. When char(F) = 2 the Plücker embedding of Q 2 has dimension N 2 ; see [7]. So gr(Q 2 ) ≥ er(Q 2 ) ≥ N 2 .…”