Invariant structure of the hierarchy theory of fractional quantum Hall states with spin

We describe the invariant structure common to Abelian fractional quantum Hall effect systems with spin. It appears in a generalization of the lattice description of the polarized hierarchy that encompasses both partially polarized and unpolarized ground-state systems. We formulate, using the spin-charge decomposition, conditions that should be satisfied so that the description is SU(2) invariant. In the case of the spin-singlet hierarchy construction, we find that there are as many SU(2) symmetries as there are levels in the construction. Various formalisms used before for hierarchies (field-theoretic, algebraic, and wave functions) are also used to show the existence of a spin and charge lattice for the systems with spin. The ``gluing'' of the charge and spin degrees of freedom in their bulk is described by the gluing theory of lattices. The low-energy field theories and corresponding quantum Hall lattices should serve as a starting point for the discussion of the stability of these systems.