  We study the nature of the multicritical point in the three-dimensional
O(3)+O(2) symmetric Landau-Ginzburg-Wilson theory, which describes the
competition of two order parameters that are O(3) and O(2) symmetric,
respectively. This study is relevant for the SO(5) theory of high-Tc
superconductors, which predicts the existence of a multicritical point in the
temperature-doping phase diagram, where the antiferromagnetic and
superconducting transition lines meet.
  We investigate whether the O(3)+O(2) symmetry gets effectively enlarged to
O(5) approaching the multicritical point. For this purpose, we study the
stability of the O(5) fixed point. By means of a Monte Carlo simulation, we
show that the O(5) fixed point is unstable with respect to the spin-4 quartic
perturbation with the crossover exponent $\phi_{4,4}=0.180(15)$, in substantial
agreement with recent field-theoretical results. This estimate is much larger
than the one-loop $\epsilon$-expansion estimate $\phi_{4,4}=1/26$, which has
often been used in the literature to discuss the multicritical behavior within
the SO(5) theory. Therefore, no symmetry enlargement is generically expected at
the multicritical transition.
  We also perform a five-loop field-theoretical analysis of the
renormalization-group flow. It shows that bicritical systems are not in the
attraction domain of the stable decoupled fixed point. Thus, in these
systems--high-Tc cuprates should belong to this class--the multicritical point
corresponds to a first-order transition.
