  We study quantum effects in a spin-3/2 antiferromagnet on the pyrochlore
lattice in an external magnetic field, focusing on the vicinity of a plateau in
the magnetization at half the saturation value, observed in CdCr$_2$O$_4$, and
HgCr$_2$O$_4$. Our theory, based on quantum fluctuations, predicts the
existence of a symmetry-broken state on the plateau, even with only
nearest-neighbor microscopic exchange. This symmetry broken state consists of a
particular arrangement of spins polarized parallel and antiparallel to the
field in a 3:1 ratio on each tetrahedron. It quadruples the lattice unit cell,
and reduces the space group from $Fd\bar{3}m$ to $P4_332$. We also predict that
for fields just above the plateau, the low temperature phase has transverse
spin order, describable as a Bose-Einstein condensate of magnons. Other
comparisons to and suggestions for experiments are discussed.
