  The possible discovery of $s_\pm$ superconducting gaps in the moderately
correlated iron-based superconductors has raised the question of how to
properly treat $s_\pm$ gaps in strongly correlated superconductors. Unlike the
d-wave cuprates, the Coulomb repulsion does not vanish by symmetry, and a
careful treatment is essential. Thus far, only the weak correlation approaches
have included this Coulomb pseudopotential, so here we introduce a symplectic N
treatment of the t-J model that incorporates the strong Coulomb repulsion
through the complete elimination of on-site pairing. Through a proper extension
of time-reversal symmetry to the large N limit, symplectic-N is the first
superconducting large N solution of the t-J model. For d-wave superconductors,
the previous uncontrolled mean field solutions are reproduced, while for
$s_\pm$ superconductors, the SU(2) constraint enforcing single occupancy acts
as a pair chemical potential adjusting the location of the gap nodes. This
adjustment can capture the wide variety of gaps proposed for the iron based
superconductors: line and point nodes, as well as two different, but related
full gaps on different Fermi surfaces.
