  We present exact explicit analytical results describing the exact ground
state of four electrons in a two dimensional square Hubbard cluster containing
16 sites taken with periodic boundary conditions. The presented procedure,
which works for arbitrary even particle number and lattice sites, is based on
explicitly given symmetry adapted base vectors constructed in r-space. The
Hamiltonian acting on these states generates a closed system of 85 linear
equations providing by its minimum eigenvalue the exact ground state of the
system. The presented results, described with the aim to generate further
creative developments, not only show how the ground state can be exactly
obtained and what kind of contributions enter in its construction, but
emphasize further characteristics of the spectrum. On this line i) possible
explications are found regarding why weak coupling expansions often provide a
good approximation for the Hubbard model at intermediate couplings, or ii)
explicitly given low lying energy states of the kinetic energy, avoiding double
occupancy, suggest new roots for pairing mechanism attracting decrease in the
kinetic energy, as emphasized by kinetic energy driven superconductivity
theories.
