  We use the $C_{4v}$ symmetry group of the 4-site Hubbard model to construct a
ground state variational wave function of two- and four interacting electrons.
In the limit $U\rightarrow 0$, ground state energies of the two- and four
interacting electrons system is of the order $-4t$. The variational wave
function of the four interacting electrons obtained using the $B_1$ irreducible
representation is valid for on-site Coulomb repulsion, while the one obtained
using the $A_1$ representation is valid for negative values of the Coulomb
interaction. The system exhibits antiferromagnetic correlations.
