  To understand the role of electronic nematic order in the interplay between
s- and d-wave particle-particle or particle-hole condensate states, relations
between various s- and d-wave order parameters are studied. We find that the
nematic operator transforms two independent six-dimensional vectors. The d-wave
superconducting, d-density wave, and antiferromagnetic orders are organized
into one vector, and the s-wave superconducting, charge density wave, and
spin-triplet d-density wave orders into the other vector. Each vector acts as a
superspin and transforms under the action of SO(6) where charge, spin, eta- and
pi-pairing, spin-triplet nematic operators satisfy the SO(6) Lie algebra.
Electronic nematic order is not a part of the SO(6) group. It commutes with all
15 generators. Our findings imply that nematic order does not affect the
competition among the order parameters within the same superspin, while it
strongly interferes the interplay between two order parameters that belong to
different superspins. For example, nematicity allows a linear coupling between
d- and s-wave superconducting order parameters which modifies the
superconducting transition temperature. A generalized Ginzburg-Landau theory
and further physical implications are discussed.
