  We prove that the centralizer algebras of the symplectic and orthogonal group
acting on tensor space are cellular algebras over the integers. We do this by
providing an axiomatic framework for studying quotient towers for towers of
diagram algebras. Our framework covers the centralizer algebras of the general
linear group acting on mixed tensor space as well. We also provide descriptions
of the kernels of the homomorphisms from the diagrammatic versions of the
algebras to the centralizer algebras on tensor space, which constitute versions
of the second fundamental theorem of invariant theory.
