I am currently self-studying representation theory of groups and I would like to verify something that I am thinking about:
A group can have only one representation on a given representation space $\mathcal{H}_{\pi}$. If you change this space, there will be another (unique) representation. I am thinking this since the action of the group is described through this representation, and it wouldn't make sense for a group to have two "different" actions on the same space.
So, whenever we talk about a group having different representations, we mean that there are representations of $G$ on a variety of different Hilbert Spaces (using the term 'different' with respect to isomorphism of spaces).
Can anyone verify my reasoning?