  The d-wave vortex lattice state is studied within the framework of
Bogoliubov-de Gennes (BdG) mean field theory. We allow antiferromagnetic (AFM)
order to develop self-consistently along with d-wave singlet superconducting
(dSC) order in response to an external magnetic field that generates vortices.
The resulting AFM order has strong peaks at the vortex centers, and changes
sign, creating domain walls along lines where $\nabla \times j_s \approx 0$.
The length scale for decay of this AFM order is found to be much larger than
the bare d-wave coherence length, $\xi$. Coexistence of dSC and AFM order in
this system is shown to induce $\pi$-triplet superconducting order. Competition
between different orders is found to suppress the local density of states at
the vortex center and comparison to recent experimental findings is discussed.
