Er-doped ${\mathrm{Al}}_2{\mathrm{O}}_3$ is a promising host for telecom-band integrated photonics. Here we combine ab initio calculations with a symmetry-resolved analysis to elucidate substitutional Er on the Al site $\left({\mathrm{Er}}_{\mathrm{Al}}\right)$ in $\alpha \text{−}{\mathrm{Al}}_2{\mathrm{O}}_3$. First-principles relaxations confirm the structural stability of ${\mathrm{Er}}_{\mathrm{Al}}$. We then use the local trigonal crystal-field symmetry to classify the Er-derived impurity levels by irreducible representations and to derive polarization-resolved electric-dipole selection rules, explicitly identifying the symmetry-allowed $f-d$ hybridization channels. Kubo-Greenwood absorption spectra computed from Kohn-Sham states quantitatively corroborate these symmetry predictions. Furthermore, we connect the calculated intra-$4f$ line strengths to Judd-Ofelt theory, clarifying the role of $4f-5d$ admixture in enabling optical activity. Notably, we predict a characteristic absorption near 1.47 $\mu \mathrm{m}$ (telecom band), relevant for on-chip amplification and emission. To our knowledge, a symmetry-resolved first-principles treatment of Er:${\mathrm{Al}}_2{\mathrm{O}}_3$ with an explicit Judd-Ofelt interpretation has not been reported, providing a transferable framework for tailoring rare-earth dopants in wide-band-gap oxides for integrated photonics. Our results for the optical spectra are in good agreement with experimental data. The resulting symmetry-based selection rules translate directly to polarization-dependent coupling in ${\mathrm{Al}}_2{\mathrm{O}}_3$ integrated photonic waveguides and resonators, enabling device-level design of TE/TM-mode interaction with Er emitters in the 1.5-$\mu \mathrm{m}$ telecom band.

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• Dopants
• Electronic structure
• Optical materials & elements
• Rare-earth doped crystals
• Density functional calculations