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).
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?