Two forms related to the symplectic dual polar space in odd characteristic

Abstract

Let V be a 2n-dimensional vector space over a field ${\mathbb{F}}$ equipped with a non-degenerate alternating form ξ. Let ${\mathcal{G}_n}$ be the n-grassmannian of PG(V) and Δ _n the dual of the polar space Δ associated to ξ. Then ${\mathcal{G}_n}$ and Δ _n are naturally embedded in the vector space ${W_n=\wedge^nV}$ and ${V_n\subseteq W_n}$ respectively, where ${\dim(W_n)=\binom{2n}{n}}$ and ${\dim(V_n)= \binom{2n}{n}-\binom{2n}{n-2}}$. The spaces W _n and V _n can be regarded as modules for the symplectic group ${Sp(2n, \mathbb{F})}$. If ${{\rm char}(\mathbb{F})\not= 2}$, we will define two forms α and β of W _n which coincide on V _n and we will investigate the relation between these two forms and the collineation of W _n naturally induced by ξ. We will obtain a description of the module W _n in terms of the two subspaces of W _n where the linear functionals induced by α and β are equal and respectively opposite.