Physicists will say that a certain system has $G$ symmetry, where $G$ is some group, such as $SU(2)$ or $S_3$ or whatever. To show that this is the case, they will conjure up an explicit representation $\rho_G$ of that group and show that the equations of motions—or the action, or whatever—are still the same. But a group is more general than a specific representation of that group, so conflating the two seems wrong.
So what does "the system has $G$ symmetry" mean?
- I don't think this can mean "There exists a representation $\rho_G$ of $G$ that is a symmetry of the system," since this is trivially true for all $G$.
- I suppose it could mean "For all representations $\rho_G$ of $G$ on the system's vector space $V$, $\rho_G$ is a symmetry." If this is so, I've never seen this much stronger statement actually shown, but maybe I'm missing something obvious.
- Knowing my colleagues, it could just mean "There is a specific representations $\rho_G$ of $G$ that is a symmetry. For cultural and linguistic reasons, we will just forget the representation information, which you can figure out on your own."
- Something else entirely?