  Low-energy transport in quantum Hall states is carried through edge modes,
and is dictated by bulk topological invariants and possibly microscopic
Boltzmann kinetics at the edge. Here we show how the presence or breaking of
symmetries of the edge Hamiltonian underlie transport properties, specifically
d.c. conductance and noise. We demonstrate this through the analysis of
hole-conjugate states of the quantum Hall effect, specifically the $\nu=2/3$
case in a quantum point-contact (QPC) geometry. We identify two symmetries, a
continuous $SU(3)$ and a discrete $Z_3$, whose presence or absence (different
symmetry scenarios) dictate qualitatively different types of behavior of
conductance and shot noise. While recent measurements are consistent with one
of these symmetry scenarios, others can be realized in future experiments.
