On the Grassmann modules for the symplectic groups

Let V be a 2n-dimensional vector space (n⩾1) over a field K equipped with a nondegenerate alternating bilinear form f, and let G≅Sp(2n,K) denote the group of isometries of (V,f). For every k∈{1,…,n}, there exists a natural representation of G on the subspace Wk of k⋀V generated by all vectors such that is totally isotropic with respect to f. With the aid of linear algebra, we prove some properties of this representation. In particular, we determine a necessary and sufficient condition for the representation to be irreducible and characterize the largest proper G-submodule. These facts allow us to determine when the Grassmann embedding of the symplectic dual polar space DW(2n−1,K) is isomorphic to its minimal full polarized embedding.