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Is it fair to say that a group representation is a particular case of a group action, in the following sense? We have the definition of group action:

A left action of a group $G$ on a set $X$ is a homomorphism $\varphi$ from the group $G$ to the group of transformations on $X$, $\varphi: G \to$ Transf($X$).

Then the definition of a group representation on a vector space (is a vector space the most general structure on which a representation is defined?) is given by

A group representation is a group action on a vector space $V$ for which the image of the action is a subset of the linear transformations on $V$: $\varphi(G) \subset \mathcal{L}(V) \subset$ Transf($V$).

Thus my questions are (1) is this definition of a representation in terms of an action correct and (2) is a vector space the most general structure on which a representation can be defined?

Edit: By group of transformations I mean the group of bijections on $X$; I'm not sure if this terminology is conventional or not.

Alp Uzman
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EE18
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  • This group is the symmetric group ${\rm Sym}(X)$. – Dietrich Burde Mar 25 '23 at 14:26
  • Hello, yes you can indeed define a representation as a group action on a vector space respecting the vector space structure. The most general definition of group action/representation would be going through category theory. Constructing the category BG from a group G. And then a "representation" in the category D, is any functor $F: BG\to D$ – DevVorb Mar 25 '23 at 14:54

1 Answers1

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Is it fair to say that a group representation is a particular case of a group action, in the following sense? We have the definition of group action: A left action of a group $G$ on a set $X$ is a homomorphism $\varphi$ from the group $G$ to the group of transformations on $X$, $\varphi: G \to$ Transf($X$).

Yes; to say the least it would be fair to expect that a representation involves a family of transformations whose parameters satisfy an algebraic relation of sorts.

Then the definition of a group representation on a vector space (is a vector space the most general structure on which a representation is defined?) is given by A group representation is a group action on a vector space $V$ for which the image of the action is a subset of the linear transformations on $V$: $\varphi(G) \subset \mathcal{L}(V) \subset$ Transf($V$).

The answer to this question is somewhat subjective and depends on the context. For some a "representation" of a group is by linear transformations, for some a "representation" is always not merely linear but indeed preserves some additional prespecified structure on the phase space $V$ (e.g. if $V$ has a Hilbert space structure additionally then the representation may be expected to be by unitary linear transformations). On the other end of the spectrum for some a "representation" is an action on a set $X$ by transformations each of which preserves some further prespecified structure on $X$. At times representations in this broader sense are referred to as "non-linear representations".

It should be noted that in any context it is also important to keep in mind what it would mean for two "representations" to be equivalent. See also How a group represents the passage of time? and Examples of conjugate-like structures across mathematics .

Alp Uzman
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