  We study the behavior of the antiferromagnetic RP$^2$ model in $d=3$. The
vacuum structure is analyzed in the critical and low temperature regions,
paying special attention to the spontaneous symmetry breaking pattern. Near the
critical point we observe a full breakdown of the O(3) symmetry of the action.
Several methods for computing critical exponents are compared. We conclude that
the most solid determination is obtained using a measure of the correlation
length. Corrections-to-scaling are parameterized, yielding a very accurate
determination of the critical coupling and a 5\% error measure of the related
exponent. This is used to estimate the systematic errors due to finite-size
effects.
