Shell model calculations have been performed for low-excitation states in the Zr isotopes between ${}^{90}\mathrm{Zr}$ and ${}^{96}\mathrm{Zr}$ with an emphasis on the $g$ factors and electromagnetic decay rates for the lowest $2^{+}$ and $3^{-}$ states. Overall the $2^{+}$ states are reasonably well described. In contrast, the $3^{-}$ states present a puzzle because the measured $g$ factors imply a single-particle configuration whereas the experimental $E3$ transition rates imply collective structures that cannot be explained by shell model calculations. A consistent description of the $3^{-}$ states in ${}^{90}\mathrm{Zr}$ and ${}^{96}\mathrm{Zr}$ is sought in terms of coupling between the single-particle structure and a collective octupole vibration.

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