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

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