  We propose an experiment to look for possible small violations of the
symmetrization postulate of Quantum Mechanics, in systems composed by three
identical particles. Such violations could be detected by investigating the
population of particular roto-vibrational states of symmetrical molecules
containing three identical nuclei. We discuss the symmetry properties of such
states, and the implications of the symmetrization postulate and of the
spin-statistics. A high sensitivity spectroscopic investigation on simple
molecules such as SO3, BH3 and NH3 could lead to the first test of the
symmetrization postulate for spin-0 and spin-1/2 nuclei.
