it is not very large so that accurate numerical calculations (in the qualitative sense) and detailed analysis can still be performed. From a previous study by a number of authors$^{13-18}$ , it is believed that a general picture of dots would consist of a core surrounding by a ring. It would be interesting to see,  when N is larger, how the details of the core-ring structure would be and how these structures are affected by symmetry . Such a study would exhibit further insight of many-body physics.

In what follows, the 13-body Schrödinger equation is solved  via an exact diagonalization of the Hamiltonian, the accuracy has been evaluated. The underlying dynamical and symmetry background has been studied.  Favorable structures for each state have been suggested based on symmetry consideration. The eigenwavefunctions have been analyzed in detail to exhibit how the electronic structures are affected by symmetry. The appearance of magic numbers is discussed.

    2, HAMILTONIAN AND THE APPROACH

Let the electrons be fully polarized (therefore the spin-part can be neglected and the spatial wave functions are totally antisymmetric) , and confined in a 2-dimensional plane by a parabolic confinement. The Hamiltonian reads

               $H=T+U\qquad \qquad (1.1)$

$\qquad T=-\sum\limits_{j=1}^{N}\frac{\hbar ^{2}}{2m^{\ast }}\nabla
_{j}^{2}\qquad 