Base subsets of symplectic Grassmannians

Pankov, Mark

Journal of Algebraic Combinatorics, Tome 26 (2007) no. 2, pp. 143-159.

Résumé

Summary: Let $V$ and $V^{\prime}$ be $2 n$-dimensional vector spaces over fields $F$ and $F^{\prime}$. Let also $\Omega : V\times V\rightarrow F$ and $\Omega ^{\prime}: V$$^{\prime}$$\times V$$^{\prime}$$\rightarrow F$$^{\prime}$ be non-degenerate symplectic forms. Denote by $\Pi $and $\Pi ^{\prime}$ the associated ($2 n - 1$)-dimensional projective spaces. The sets of $k$-dimensional totally isotropic subspaces of $\Pi $and $\Pi ^{\prime}$ will be denoted by $G _{ k} {\mathcal G}$_k and $G \textcent _{ k} {\mathcal G}$'_k, respectively. Apartments of the associated buildings intersect $G _{ k} {\mathcal G}$_k and $G \textcent _{ k} {\mathcal G}$'_k by so-called base subsets. We show that every mapping of $G _{ k} {\mathcal G}$_k to $G \textcent _{ k} {\mathcal G}$'_k sending base subsets to base subsets is induced by a symplectic embedding of $\Pi $to $\Pi ^{\prime}$.

Détail

Keywords: keywords Tits building, symplectic grassmannians, base subsets