The irreducible representations of the Lie algebra ${frak su}$(3) describe rotational bands in the context of the nuclear shell and interacting boson models. The density matrices associated with ${frak su}$(3) provide an alternative theoretical framework for obtaining these bands. The ${frak su}$(3) density matrix formulation is mathematically simpler than representation theory, yet it yields similar results. Bands are solutions to a system of polynomial equations defined by the quadratic and cubic ${frak su}$(3) Casimirs. Analytic solutions are found in many physically important cases including rotation about principal axes and spheroids. Numerical solutions are reported in other cases including tilted rotors. The physics of rotational bands is more transparent in the presented formalism. In representation theory bands terminate because the space is finite-dimensional. In ${frak su}$(3) density matrix theory bands terminate when faster rotation produces a spheroid rotating around its symmetry axis.
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