Multi-channel collective dissipation via the symmetric irreducible representation of SU(4)
We specialize Agarwal's multi-level collective spontaneous-emission formalism to the four-level case by formulating it in the fully symmetric \SU(4) representation of $N$ identical atoms. In the irreducible representation $(N,0,0)$, the occupation-number basis forms a tetrahedral weight lattice on which the six embedded $\mathfrak{su}(2)$ transition subalgebras act as ladder operators. From these algebraic factors we obtain a compact Pauli-type population-rate equation and a closed-form expression for the total emitted intensity that apply to any combination of open dipole channels. The formalism is then specialized to the seven dipole-allowed four-level topologies -- tripod, inverted tripod, Y, inverted Y, double-$Λ$, closed cascade, and diamond -- and the resulting rate equations are solved numerically for atom numbers up to $N=50$. In every case the emitted intensity develops a delayed cooperative burst whose peak height obeys a power law $I_{\mathrm{peak}}=aN^{p}$ with topology-dependent parameters $(a,p)$; the fitted exponents lie in the range $1.81\lesssim p\lesssim 1.92$, indicating a superlinear. The \SU(4) tetrahedral flow and the seven configuration-dependent transients together provide a unified geometric picture of multi-channel collective dissipation in four-level atomic ensembles.