  Consider a high-energy parton showering as it traverses a QCD medium such as
a quark-gluon plasma. Interference effects between successive splittings in the
shower are potentially very important but have so far been calculated (even in
idealized theoretical situations) only in soft emission or large-$N_{\rm c}$
limits, where $N_{\rm c}$ is the number of quark colors. In this paper, we show
how one may remove the assumption of large $N_{\rm c}$ and so begin
investigation of $N_{\rm c}{=}3$ without soft-emission approximations. Treating
finite $N_{\rm c}$ requires (i) classifying different ways that four gluons can
form a color singlet and (ii) calculating medium-induced transitions between
those singlets, for which we find application of results for the generalization
of Wigner 6-$j$ symbols from angular momentum to SU($N_{\rm c}$). Throughout,
we make use of the multiple scattering ($\hat q$) approximation for high-energy
partons crossing quark-gluon plasmas, and we find that this approximation is
self-consistent only if the transverse-momentum diffusion parameter $\hat q$
for different color representations satisfies Casimir scaling (even for
strongly-coupled, and not just weakly-coupled, quark-gluon plasmas). We also
find that results for $N_{\rm c}{=}3$ depend, mathematically, on being able to
calculate the propagator for a coupled non-relativistic quantum harmonic
oscillator problem in which the spring constants are operators acting on a
5-dimensional Hilbert space of internal color states. Those spring constants
are represented by constant $5{\times}5$ matrices, which we explicitly
construct. We are unaware of any closed form solution for this type of harmonic
oscillator problem, and we discuss prospects for using numerical evaluation.
