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

Let V be the Weyl module of dimension (2n n) - (2n n-2) for the symplectic group Sp(2n, F) whose highest weight is the nth fundamental dominant weight. The module V affords the grassmann embedding of the symplectic dual polar space DW (2n - 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 I. Here we provide a geometric description for Rand study some relations between Rand other objects associated with the grassmann embedding. (C) 2009 Elsevier B.V. All rights reserved.