The cellular second fundamental theorem of invariant theory for classical groups

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.