Homology representations of GL(n,q) from Grassmannians in cross-characteristics

Abstract

Let F* be the field of q elements and let P(n,q) denote 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) for a coefficient field F of positive characteristic co-prime to q. As GL(n,q)-representations these modules are obtained from the permutation action of GL(n,q) on the set of subspaces of F*. We prove a branching rule for the H^{n}_{k,i} and use this to determine the homology representations completely. Results include a duality theorem, the characterisation of H^{n}_{k,i} through the standard irreducibles of GL(n,q) over F and applications.