The p-rank of the incidence matrix of intersecting linear subspaces
Let V be a vector space of dimension n + 1 over a field of p(t) elements. A d-dimensional subspace and an e-dimensional subspace are considered to be incident if their intersection is not the zero subspace. The rank of these incidence matrices, modulo p, are computed for all n, d, e and t. This result generalizes the well-known formula of Hamada for the incidence matrices between points and subspaces of given dimensions in a finite projective space. A generating function for these ranks as t varies, keeping n, d and e fixed, is also given. In the special case where the dimensions are complementary, i.e., d + e = n + 1, our formula improves previous upper bounds on the size of partial m-systems ( as defined by Shult and Thas).
p-rank incidence matrices permutation modules