Why must the reflected and refracted waves of S/P-polarized EM wave remain S/P-polarization? When deriving Snell's law and Fresnel's law, we decompose all elliptically polarized waves into S and P waves. But we assumed that S wave retains its polarization after reflected or refracted. Why this assumption is correct?
Or are S and P polarized waves always independent to each other? 
 A: A pure S or pure P polarization is maintained if the medium is not partially absorbing and does not have the kind of crystalline structure or stresses that can result in polarization rotation (that is, the medium is not birefringent or optically active).
The reason why a pure S polarization or pure P polarization is maintained in an ordinary material such as un-stressed glass comes down to the way the wave is transported through the material. The incoming wave creates oscillating dipoles throughout the material; these are driven to oscillate by the wave, and they also have internal forces from the material. If the material is isotropic then the internal forces act simply like restoring forces with no preferred direction, and therefore the net result is that the dipoles oscillate in the same direction as the electric field of the incoming wave. Therefore all the waves that result from the combination of these dipoles and the incoming wave are also polarized in that same direction.
In the case of linear polarization in some other direction, such as 45 degrees to the plane of incidence, the field inside the medium will oscillate in a direction such that it matches the boundary conditions, i.e. the sum of incident and reflected fields, and this will not necessarily be an oscillation at the same angle to the plane of incidence (e.g. 45 degrees) as the incident field. In fact it usually will not be. But still the dipoles inside the medium oscillate in the same direction as the field inside the medium. To find the polarization angle in this case, whether of the transmitted or reflected beam, one convenient way is to decompose the incident field into S and P and find the transmission and reflection coefficients for each. [Thanks to Ofek Gillon for prompting me to add this paragraph].
