  The consequences of the attractive, short-range nucleon-nucleon (NN)
interaction on the wave functions of the Elliott SU(3) and the proxy-SU(3)
symmetry are discussed. The NN interaction favors the most symmetric spatial
SU(3) irreducible representation, which corresponds to the maximal spatial
overlap among the fermions. The percentage of the symmetric components out of
the total in an SU(3) wave function is introduced, through which it is found,
that no SU(3) irrep is more symmetric than the highest weight irrep for a
certain number of valence particles in a three dimensional, isotropic, harmonic
oscillator shell. The consideration of the highest weight irreps in nuclei and
in alkali metal clusters, leads to the prediction of a prolate to oblate shape
transition beyond the mid-shell region.
