Some homology representations for Grassmannians in cross-characteristics

Let $\mathbb F$ be the finite field of $q$ elements and let $\mathcal P(n,q)$ denote the projective space of dimension $n-1$ over $\mathbb F$. We construct a family $H^n_{k,i}$ of combinatorial homology modules associated to $\mathcal P(n,q)$ for coefficient fields of positive characteristic co-prime to $q$. As $F\mathrm{GL}(n,q)$-representations these modules are obtained from the permutation action of $\mathrm{GL}(n,q)$ on the Grassmannians of $\mathbb F^n$. We prove a branching rule for $H^n_{k,i}$ and use this to determine the homology representations completely. Our results include a duality theorem and the characterisation of $H^n_{k,i}$ through the standard irreducibles of $\mathrm{GL}(n,q)$ over $F$.