Random interactions in nuclei and extension of $0^+$ dominance in ground   states to irreps of group symmetries
  Random one plus two-body hamiltonians invariant with respect to $O({\cal N}_1) \oplus O({\cal N}_2)$ symmetry in the group-subgroup chains $U({\cal N}) \supset U({\cal N}_1) \oplus U({\cal N}_2) \supset O({\cal N}_1) \oplus O({\cal N}_2)$ and $U({\cal N}) \supset O({\cal N}) \supset O({\cal N}_1) \oplus O({\cal N}_2)$ chains of a variety of interacting boson models are used to investigate the probability of occurrence of a given $(\omega_1 \omega_2)$ irreducible representation (irrep) to be the ground state in even-even nuclei; $[\omega_1]$ and $[\omega_2]$ are symmetric irreps of $O({\cal N}_1)$ and $O({\cal N}_2)$ respectively. Numerical results obtained for ${\cal N}_1 \geq 3, {\cal N}_2=1$ and ${\cal N}_1, {\cal N}_2 \geq 3$ situations are well explained by an extended Hartree-Bose mean-field analysis. 
V.K.B. Kota