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I know that both are totally symmetric and that the '$1$', according to the Mulliken table, refers to symmetry around the $C_2$ axis but I was reading some papers and found that modes were sometimes referred to as $A_{1g}$ and sometimes referred to $A_{g}$, as shown below (difference is between undistorted and distorted unit cell).

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Is $A_{g}$ equivalent to $A_{1g}$ here because the Raman tensors are identical for both (both are a 3 by 3 maxtrix with a leading diagonal of $a, a, b$) and yet the names are slightly different?

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To answer your question of A$_{1g}$ and A$_g$ are the same? No, they are not, because they correspond to irreducible representations of different symmetry point groups that differ in the allowed symmetry operations. However, they correspond to the same vibrational mode, since the quadratic functions associated with both these modes are same. If you see the Character Table for D$_{6h}$ and C$_{6h}$, you would see that A$_{1g}$ corresponds to the fully symmetric mode of vibration, and there are other modes of A, B and E types. However, once you lower the symmetry to C$_{6h}$ symmetry, you loose some elements of symmetry since C$_{6h}$ is a subgroup of D$_{6h}$. This would be more clear if you see the character tables available online D$_{6h}$ and C$_{6h}$

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