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For definiteness, consider the group $SO(3)$. There is a Lie algebra $so(3)$ given by

$$ [T_a, T_b] = if_{abc}T_c $$

The generators of this algebra can be exponentiated to form the elements of $SO(3)$,

$$ \text{exp}[i\alpha_a T_a] \in SO(3) $$

There are various ways to represent the generators $T_a$ in such a way that they still obey the Lie algebra so(3). This in turn yields different representations of the Lie group, SO(3).

In this context, it seems that the representations we are talking about are specific realizations of the elements of SO(3) -- that is, 3x3 orthogonal matrices with unit determinant. But then the literature also makes statements like "the fundamental representation of SO(3) is given by the 3 component vector v, which transforms under a group element $O \in SO(3)$ via $ v \to Ov$." To my ear they are saying the representation of SO(3) is actually the vector v, and not the matrix O.

So my question is: which is it? Are the representations of a symmetry group the generators of that group, the elements of the group itself, or the objects that the group acts upon?

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  • $\begingroup$ A little forbearance can get you a long way. A linear map of vectors, or the matrix realizing it, or the logarithmic quiddity thereof can hardly confuse the well-meaning, non-pedantic reader, in any crucial way.... $\endgroup$
    – Cosmas Zachos
    Apr 7, 2018 at 19:59
  • $\begingroup$ I think the OP should withdraw it. $\endgroup$
    – Cosmas Zachos
    Apr 7, 2018 at 19:59
  • $\begingroup$ Different people use the word differently. Mathematicians usually say the representation is the map, as @AccidentalFourierTransform pointed out. But usually when you're doing a specific problem, the representation is fixed from the start (i.e. $U(g)$ for a rotation $g \in SO(3)$ is always just physical spatial rotation, you would never consider a different representation $U'(g)$) and we care much much more about how the Hilbert space decomposes. So in this case we talk about representations as vector spaces. $\endgroup$
    – knzhou
    Apr 7, 2018 at 20:07
  • $\begingroup$ A representation of a group $G$ is a pair $(V,\rho)$ where $V$ is a vector space and $\rho : G\to GL(V)$ is a group homomorphism. In other words: it is one particular way on which the group can act upon elements of a vector space in a linear way. It is common that sometimes people call $V$ itself the representation, but the homomorphism is actually the important part. See, e.g., the book "Representation Theory: A First Course" by Fulton and Harris. $\endgroup$
    – Gold
    Apr 7, 2018 at 20:11

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