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In Fulton and Harris's book Representation Theory: A First Course, they define a representation of a finite group on $V$ in Lecture 1. Then they say that the representation gives $V$ the structure of a $G$-module. How do we understand it? $G$-module? I do not think $G$ is a ring. I am interested in how to interpret this "vague" statement?

The definition they give:

A representation of a finite group $G$ on a finite-dimensional complex vector space $V$ is a homomorphism of $\rho: G \rightarrow GL(V)$ of $G$ to the group of automorphisms of $V$; we say that such a map gives $V$ the structure of a $G$-module.

Conclusion: As explained below, the $G$-module that they refer to is fact a $\mathbb{C}[G]$-module. Some textbooks, e.g. Representations of Compact Lie Groups by Br$\mathrm{\ddot{o}}$cker and Dieck, refer to it just in that way with abuse of notation.

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    $\begingroup$ It would not hurt, and it would make your question useful to people who do not have the book (they do exist!), to spell out the definitons of representation and of $G$-module that the book use. $\endgroup$
    – Mariano Suárez-Álvarez
    Jun 28, 2014 at 7:29
  • $\begingroup$ @MarianoSuárez-Alvarez I did not go through their book, but this is the very first definition they give, in the usual sense. No G-module is defined in the front. I wonder whether there is an alternative definition of G-module in category theory or somewhere. $\endgroup$
    – DarKnightS
    Jun 28, 2014 at 7:36
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    $\begingroup$ Well, how do they define representation? Keep in mind that your question is impossible to make sense of if one does not have the book. For example, I am sure I could answer your question, but I do not rememeber what definition of representation they use, nor if and then how they define $G$-modules. $\endgroup$
    – Mariano Suárez-Álvarez
    Jun 28, 2014 at 7:39
  • $\begingroup$ In any case, $GL(V)$ is certainly not a ring! $\endgroup$
    – Mariano Suárez-Álvarez
    Jun 28, 2014 at 7:39
  • $\begingroup$ @MarianoSuárez-Alvarez: A representation of a finite group G on a finite-dimensional complex vector space V is a homomorphism of G to the group of Aut(V). Then it comes the statement. $\endgroup$
    – DarKnightS
    Jun 28, 2014 at 7:43

1 Answer 1

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If $G$ acts on a set $X$, we observe that there is a group homomorphism between $G$ and $Aut(X)$. Here $Aut(X)$ is the set of automorphisms of $X$ as a set, meaning only bijective mapping from $X$ to $X$.

Now if our set $X$ being acted on has linear space structure, and the action of $G$ on the linear space $V$ respects the linear structure of $X$, then we say this is a linear action. In this case we have a group homomorphism between $G$ and $Aut(V)$, but this time $Aut(V)$ consists of linear automorphisms of $V$.

Regarding the term module, the group action $G\times V\to V$ induces a module over the group ring $\mathbb{C}[G]\times V\to V$, which encodes all the information of the group representation. There is a functor from the category of linear modules over a group $G$ to the category of linear ring modules over the group ring $\mathbb{C}[G]$.

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    $\begingroup$ I guess it is just a matter of convention. ``$G$-module'' to me is a little bit inappropriate. $\mathbb{C}[G]$-module is certainly correct. $\mathbb{C}G$-space is what I would use to define a representation. $\endgroup$
    – DarKnightS
    Jun 28, 2014 at 8:40
  • $\begingroup$ The term $G$-module means "a module with an action of $G$ which respects the structure as a module". It is used when the field (or ring) has already been made clear previously. The term $\mathbb{C}G$-space is not one I have seen before, and I would find it unusual. Note that the term $G$-space then means the same as above with "module" replaced by "space", which is a bit tricky, since the word space is so overloaded. $\endgroup$
    – Tobias Kildetoft
    Jun 28, 2014 at 9:41
  • $\begingroup$ The notation $\mathbb{C}G$-space is used by Frank Adams in his $\it{Lectures\ on\ Lie\ Groups}$, which is a ''classic.'' $\endgroup$
    – DarKnightS
    Jun 28, 2014 at 10:05

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