The vibrational modes in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ easily induce octahedral rotations without tilting. Being on the verge of a magnetic instability, such propensity of octahedral rotation may also produce magnetic fluctuations. In this work, we analyze the long-range magnetic phase diagram incorporating such octahedral rotations and demonstrate the possibility of an altermagnetic phase in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. Using ab initio calculations, we first study single-layer ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ with octahedral rotations, obtaining an orbital-selective $g$-wave altermagnetic phase. We further provide an effective $t_{2g}$ tight-binding model, demonstrating that the $g$-wave altermagnetism is primarily a product of second- and third-nearest-neighbor interorbital hybridizations between the $\gamma z$ ($\gamma =x,y$) orbitals, but only a much longer range intraorbital hybridization in the $xy$ orbitals, establishing a strong orbital selectiveness for the altermagnetism. Notably, by replacing the $xy$ orbital with the $x^2-y^2$ orbital, a similar tight-binding model may be used to investigate the hole-doped cuprate superconductors. We then study bulk ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$, where we find the altermagnetic phase as the magnetic ground state for a range of finite octahedral rotations. In the bulk, interlayer hopping breaks some of the symmetries of the $g$-wave altermagnet, resulting in a $d_{xy}$-wave altermagnet, still with orbital selectiveness. We also include relativistic effects through spin-orbit coupling and obtain that an effective staggered Dzyaloshinskii-Moriya interaction generates weak ferromagnetism. Finally, we discuss the implications of the altermagnetic order on the intrinsic superconductivity of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. Assuming in-plane intraorbital pairing, the altermagnetism favors spin-singlet $d_{x^2-y^2}$-wave or $g$-wave pairing, or their combinations.

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• Magnetic order
• Magnetism
• Altermagnets
• Ruthenates
• Density functional calculations
• Tight-binding model