  We introduce a {\em local} formalism, in terms of eigenstates of number
operators, having well defined point symmetry, to solve the Hubbard model at
weak coupling on a $N\times N$ square lattice (for even $N$). The key concept
is that of W=0 states, that are the many-body eigenstates of the kinetic energy
with vanishing Hubbard repulsion. At half filling, the wave function
demonstrates an antiferromagnetic order, a lattice step translation being
equivalent to a spin flip. Further, we state a general theorem which allows to
find all the W=0 pairs (two-body W=0 singlet states). We show that, in special
cases, this assigns the ground state symmetries at least in the weak coupling
regime. The N=4 case is discussed in detail. To study the doped half filled
system, we enhance the Group Theory analysis of the $4 \times 4$ Hubbard model
introducing an Optimal Group which explains all the degeneracies in the
one-body and many-body spectra. We use the Optimal Group to predict the
possible ground state symmetries of the $4 \times 4$ doped antiferromagnet by
means of our general theorem and the results are in agreement with exact
diagonalization data. Then we create W=0 {\it electron} pairs over the
antiferromagnetic state. We show analitycally that the effective interaction
between the electrons of the pairs is attractive and forms bound states.
Computing the corresponding binding energy we are able to definitely predict
the exact ground state symmetry.
