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We study the pairing symmetry of Sr$_2$RuO$_4$ through the group-theoreticalapproach. We emphasize the role of pairing interaction between thequasi-one-dimensional(Q1D) $d_{xz/yz}$ and quasi-two-dimensional(Q2D) $d_{xy}$orbitals. It is found that two degenerate inter-orbitaltime-reversal-invariant(TRI) p-wave pairings, one is spin-singlet and the otherspin-triplet with out-of-plane $\bm{d}$-vector, could be the most promisingcandidates. Several important physical quantities are presented, including thenear-nodal gap structure, the unchanged out-of-plane Knight shift, and no splittransition under strain, which are consistent with the experiments. Inaddition, these p-wave pairings shed light on resolving the contradictionbetween the time-reversal breaking and reduced in-plane Knight shiftmeasurements. As the system reaches the Van Hove singularity under appliedstrain, the pairing symmetry would become a d-wave pairing mainly consisting ofinter-orbital components, which could be responsible for the strained 3$K$phase.

Inter-orbital p- and d-wave pairings between $d_{xz/yz}$ and $d_{xy}$ orbitals in Sr$_2$RuO$_4$