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

• Authors: Ilaria Cardinali;Antonio Pasini
• Venue: Designs, Codes and Cryptography
• Year: 2012

Let V be a 2n-dimensional vector space over a field $${\mathbb{F}}$$ equipped with a non-degenerate alternating form 驴. Let $${\mathcal{G}_n}$$ be the n-grassmannian of PG(V) and Δ n the dual of the polar space Δ associated to 驴. 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 驴 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 驴. 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 驴 and β are equal and respectively opposite.