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I am reading Schwartz's book "Quantum Field Theory and The Standard Model", in chapter 8, the author says "A set of objects $\psi$ that mix under a transformation group is called a representation of the group."

I am confused about this, I thought the representation is the representation of the transformation, not the set of states corresponding to those transformation. In the book, it sounds like the set of states is the representation, which doesn't make sense to me, how can a set of states (closed under the transformation) be representation of the group? Or is it just really saying the "transformation matrix" (assume we are using matrix representation) is the representation of the group, and the set of states closed under those transformation is also called the representation of that group?

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A representation $\rho$ of some Lie group $G$ is a group homomorphism $\rho:G\rightarrow GL(V)$, where $GL(V)$ is the general linear group over some vector space $V$. $V$ is often called the representation space.

It is standard parlance both in physics and in mathematics to refer to a representation space simply as a representation when the actual homomorphism in question is otherwise clear from context.

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  • $\begingroup$ Thanks for the reply, I understand it now, so when people talking about "we want unitary representation", they are talking about wanting the subset of the vector in that vector space $V$ that transform covariantly? $\endgroup$
    – Tea_de
    Oct 17, 2020 at 4:26
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    $\begingroup$ @Tea_de No, a unitary representation $\rho:G \rightarrow U(V) \subseteq GL(V)$ is a representation which takes its values in the unitary maps from $V\rightarrow V$, meaning that for all $g\in G$, $\rho(g)$ is a unitary operator. $\endgroup$
    – J. Murray
    Oct 17, 2020 at 4:29

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