  The transmission phase through a quantum dot with few electrons shows a
complex, non-universal behavior. Here we combine configuration-interaction
calculations ---treating rigorously Coulomb interaction--- and the Friedel sum
rule to provide a rationale for the experimental findings. The phase evolution
for more than two electrons is found to strongly depend on dot's shape and
electron density, whereas from one to two the phase never lapses. In the
Coulomb (Kondo) regime the phase shifts are significant fractions of pi (pi/2)
for the second and subsequent charge addition if the dot is strongly
correlated. These results are explained by the proper inclusion in the theory
of Coulomb interaction, spin, and orbital degrees of freedom.
