Group symmetries in two-body random matrix ensembles generating order out of complexity

The two-body random matrix ensembles with spin TBRE-s and in a single j shell TBRE-j introduced recently in the context of ground state structures in complex interacting particle systems, possess U(N) superset U(N/2)multiply sign in circle SU(2) and U(N)superset O(3) group symmetries, respectively, with N the number of single particle states. It is shown that both these group symmetries give rise to simplicities in the ground state structures but in different ways.