Symmetry properties and residual transport in superconducting PrOs$_4$Sb$_{12}$

We identify a three-component order parameter in the triplet channel as the most probable candidate for superconductivity in PrOs$_4 $Sb$_{12}$. Two different superconducting phases have been observed in PrOs$_4$Sb$_{12}$; the lower temperature `B phase' occupies the bulk of the phase diagram and breaks time reversal symmetry while the higher temperature `A phase' is found in a narrow region below H$_{c2} $, and possibly does not exist at all. The gap function in the A phase is unitary and has two nodes in the $[001]$ direction. In the B phase, the gap function is nonunitary and the lower branch has four cusp nodes in the $[\pm\alpha,\pm\beta,0]$ directions. The conductivity tensor, due to isotropic impurity scattering, has inequivalent diagonal components due to the off-axis nodal positions of the B phase.