  A formalism is presented for treating strongly-correlated graphene quantum
Hall states in terms of an SO(8) fermion dynamical symmetry that includes
pairing as well as particle--hole generators. The graphene SO(8) algebra is
isomorphic to an SO(8) algebra that has found broad application in nuclear
physics, albeit with physically very different generators, and exhibits a
strong formal similarity to SU(4) symmetries that have been proposed to
describe high-temperature superconductors. The well-known SU(4) symmetry of
quantum Hall ferromagnetism for single-layer graphene is recovered as one
subgroup of SO(8), but the dynamical symmetry structure associated with the
full set of SO(8) subgroup chains extends quantum Hall ferromagnetism and
allows analytical many-body solutions for a rich set of collective states
exhibiting spontaneously-broken symmetry that may be important for the
low-energy physics of graphene in strong magnetic fields. The SO(8) symmetry
permits a natural definition of generalized coherent states that correspond to
symmetry-constrained Hartree--Fock--Bogoliubov solutions, or equivalently a
microscopically-derived Ginzburg--Landau formalism, exhibiting the interplay
between competing spontaneously broken symmetries in determining the ground
state.
