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Consider a high-energy parton showering as it traverses a QCD medium such asa quark-gluon plasma. Interference effects between successive splittings in theshower are potentially very important but have so far been calculated (even inidealized 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 showhow one may remove the assumption of large $N_{\rm c}$ and so begininvestigation of $N_{\rm c}{=}3$ without soft-emission approximations. Treatingfinite $N_{\rm c}$ requires (i) classifying different ways that four gluons canform a color singlet and (ii) calculating medium-induced transitions betweenthose singlets, for which we find application of results for the generalizationof 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-energypartons crossing quark-gluon plasmas, and we find that this approximation isself-consistent only if the transverse-momentum diffusion parameter $\hat q$for different color representations satisfies Casimir scaling (even forstrongly-coupled, and not just weakly-coupled, quark-gluon plasmas). We alsofind that results for $N_{\rm c}{=}3$ depend, mathematically, on being able tocalculate the propagator for a coupled non-relativistic quantum harmonicoscillator problem in which the spring constants are operators acting on a5-dimensional Hilbert space of internal color states. Those spring constantsare represented by constant $5{\times}5$ matrices, which we explicitlyconstruct. We are unaware of any closed form solution for this type of harmonicoscillator problem, and we discuss prospects for using numerical evaluation.

The LPM effect in sequential bremsstrahlung: from large-N QCD to N=3 via the SU(N) analog of Wigner 6-j symbols