On the grassmann module of symplectic dual polar spaces of rank 4 in characteristic 3

Let V be the Weyl module of dimension 2 n n − 2 n n − 2 for the symplectic group Sp ( 2 n , F ) whose highest weight is the n th fundamental dominant weight. The module V affords the grassmann embedding of the symplectic dual polar space D W ( 2 n − 1 , F ) , therefore V is also called the grassmann module for the symplectic group. We consider the smallest case for char ( F ) odd for which V is reducible, namely n = 4 and char ( F ) = 3 . In this case the unique factor R of V has vector dimension 1. Here we provide a geometric description for R and study some relations between R and other objects associated with the grassmann embedding.