# Is there uniqueness of a group representation on a given representation space?

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?

• Hi Natasha, do you know about Schur's lemma? en.wikipedia.org/wiki/Schur%27s_lemma It's an extremely useful statement in rep theory for reasoning about these kinds of questions. Jan 25, 2021 at 22:53
• No. The simplest example is that the $2$-element group has two non-isomorphic $1$-dimensional representations (over any field of characteristic $\neq2$. Jan 26, 2021 at 1:57