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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?

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  • $\begingroup$ 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. $\endgroup$
    – Condo
    Jan 25, 2021 at 22:53
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    $\begingroup$ No. The simplest example is that the $2$-element group has two non-isomorphic $1$-dimensional representations (over any field of characteristic $\neq2$. $\endgroup$ Jan 26, 2021 at 1:57

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