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^{ n}_{ k,i} of combinatorial homology modules associated to P(n,q) over a coefficient field F field of characteristic p_{ 0}>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*^n. We prove a branching rule for H^{ n}_{ k,i} and use this rule to determine these homology representations completely. The main results are a duality theorem and the complete characterisation of H^{ n}_{ k,i} in terms of the standard irreducibles of GL(n,q) over F and applications.
Categories: math.CO math.GR
