  We use group theoretic ideas and coset space methods to deal with problems in
polarization optics of a global nature. These include the possibility of a
globally smooth phase convention for electric fields for all points on the
Poincar\'{e} sphere, and a similar possibility of real or complex bases of
transverse electric vectors for all possible propagation directions. It is
shown that these methods help in understanding some known results in an
effective manner, and in answering new questions as well. We find that apart
from the groups $SU(2)$ and $SO(3)$ which occur naturally in these problems,
the group $SU(3)$ also plays an important role.
