  Let k be an algebraically closed field of characteristic p>0. We compute the
Weyl filtration multiplicities in indecomposable tilting modules and the
decomposition numbers for the general linear group over k in terms of cap
diagrams under the assumption that p is bigger than the greatest hook length in
the partitions involved. Then we introduce and study the rational Schur functor
from a category of GL_n-modules to the category of modules for the walled
Brauer algebra. As a corollary we obtain the decomposition numbers for the
walled Brauer algebra when p is bigger than the greatest hook length in the
partitions involved. This is a sequel to an earlier paper on the symplectic
group and the Brauer algebra.
