Grassmann and Weyl embeddings of orthogonal Grassmannians

Cardinali, Ilaria ; Pasini, Antonio

Journal of Algebraic Combinatorics, Tome 38 (2013) no. 4, pp. 863-888.

Résumé

Given a non-singular quadratic form $q$ of maximal Witt index on $V := V(2n+1,\mathbb{F})$, let $\varDelta $ 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 $\varDelta _{k }$ be the $k$-grassmannian of $\varDelta $. Let $\varepsilon _{k }$ be the embedding of $\varDelta _{k }$ into PG$(\bigwedge^{k } V)$ mapping every point $\langle v _{1},v _{2},\dots ,v _{k }\rangle $ of $\varDelta _{k }$ to the point $\langle v _{1}\land v _{2}\land \dots \land v _{k }\rangle $ of PG$(\bigwedge^{k } V)$. It is known that if char$(\mathbb{F})\neq2$ then 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 char$(\mathbb{F}) = 2$ then dim$(\varepsilon_{k})={{2n+1}\choose k}-{{2n+1}\choose{k-2}}$. As a consequence, when $1$ and char$(\mathbb{F}) = 2$ the embedding $\varepsilon _{k }$ 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 $\varepsilon _{k }$ is universal.

Détail

Classification : 14M15, 51A50, 51A45, 51E24, 51M35
Keywords: orthogonal Grassmannian, Grassmann embedding, Weyl embedding, universal embedding