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 Sp(2n)(2) is lambda(n) = (2(n) + l)(2(n-1) + l)/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 tired 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 Sp(2n)(2) of cardinality lambda(n). (C) 2000 Academic Press.