Incidence Homology of Finite Projective Spaces

Let F be the finite field of q elements and let P(n, q) be the projective space of dimension n−1 over F. We construct a family H k,i of combinatorial homology modules associated to P(n, q) over a coefficient field F field of characteristic p0 > 0 co-prime to q. As FGL(n, q) -representations the modules are obtained from the permutation action of GL(n, q) on the subspaces of F. We prove a branching rule for H k,i and use this rule to determine these homology representations completely. The main results are a duality theorem and the characterisation of H k,i in terms of the standard irreducibles of GL(n, q) over F.