Comparative study of the 1-2 exchange symmetries in neutrino frameworks with global and local validities

A new picture ``one-resonance - one-symmetry'' has been proposed recently to reveal nature of the reparametrization symmetry in neutrino oscillation in matter and where it resides: Symmetry of $i \leftrightarrow j$ state-exchange type exists at around a resonance with $i$ and $j$ being the states which participate in the level crossing. Consistently, the 1-2 and 1-3 state exchange symmetries are identified at around the solar and atmospheric resonances, respectively, in the locally-valid frameworks. On the other hand, the Denton {\it et al.} (DMP) perturbation theory, a globally-valid framework, has the 1-2 exchange symmetry which is akin to the one in the aforementioned solar-resonance perturbation (SRP) theory. In our picture, the symmetry must be associated with the resonance, not the framework, and if so these two 1-2 symmetries must be identical to each other. We conduct a comparative study of the 1-2 symmetries possessed by SRP and DMP to confirm their identity. An almost identity is verified, but in a highly nontrivial way.