On the Universal Embedding of Dual Polar Spaces of Type Sp2n(2)

It has been conjectured by A. E. Brouwer that the dimension of the universal embedding module of a dual polar space of type Sp2n(2) is λ(n)=(2n+1)(2n−1+1)/3. Following a point stabilizer approach of A. A. Ivanov and M. K. Bardoe, we investigate the dimensions of certain quotients of permutation modules for SLn(2) on subspaces of a fixed vector space of dimension n. This is accomplished by studying the nullity of associated incidence matrices over GF(2). In the process we provide evidence of a generating set for the dual polar space of type Sp2n(2) of cardinality λ(n).