# Doubt: "A group representation is exactly like a module over the group ring"

It is traditional to say that a representation of a group $$G$$ over a field $$F$$ is "exactly like" a module over the group ring $$F[G]$$.

I think it is inaccurate. I think a module over $$F[G]$$ encodes more than a representation of $$G$$. I will give an example of two modules over $$F[G]$$ giving rise to the same group representation of $$G$$.

I will give the simplest example:

Let $$G$$ be the trivial group of order $$1$$, and let $$F=\mathbb{C}$$. I can give the abelian group $$M=\mathbb{C}$$ a structure of a $$\mathbb{C}[G]$$-module in different ways. I can let each $$z\cdot 1_G\in\mathbb{C}$$ act on $$m\in M$$ either by $$(z\cdot 1_G)\cdot m=zm$$, or by $$(z\cdot 1_G)\cdot m=\overline{z}m$$.

Is it true to say that there is a bijective correspondence between the following collections?

1. The collection of pairs $$(\rho,\theta)$$, where $$\rho$$ is a group representation of $$G$$ over $$F$$, and $$\theta$$ is a field endomorphism of $$F$$, and
2. The collection of modules over $$F[G]$$
• Letting a complex number act by scalar multiplication by its conjugate is not "legal", that is, it is not $\mathbb C$-linear. That is a fatal objection if the representations you wish to consider are to be on complex vector spaces. Nov 15, 2013 at 0:01
• @paulgarrett: But does it define a $\mathbb{C}[G]$-module?
– Gils
Nov 15, 2013 at 0:02
• In a "technical" sense, it does not define a $\mathbb C[G]$-module, because the usual convention (!) for the latter is that $\mathbb C$ acts $\mathbb C$-linearly. That is, one is not "at liberty" to fool around with the action of complex numbers. I do agree, this might be under-emphasized in some presentations, all the worse that one might reasonably think of the complex numbers as an "optional" extension of the "more natural" real numbers. Nov 15, 2013 at 0:06
• @paulgarrett: I understand. Thanks.
– Gils
Nov 15, 2013 at 0:10

It is possible to define a $\mathbb{C}[G]$-module structure like this, but the structure is not compatible with the $\mathbb{C}$-vector space structure on $M$. When you define a representation, you start with a vector space, so the action of the base field is prescribed.

So these two module structures are two different representation:

• the usual one;
• the one that starts with a different vector space over $\mathbb{C}$, that you might call $\overline M$, and the action you described.

As for your more general question: you have two categories, the category of $\mathbb{C}[G]$-modules and the category of representations of $G$ (where objects are pairs $(V, \rho)$, $V$ a vector space, $\rho : G \to End(V)$). Then these two categories are equivalent.

• Thanks. I upvoted, but did not accept, because this does not address my question in the gray box.
– Gils
Nov 15, 2013 at 0:12
• FWIW your question doesn't really makes sense. Your are asking for a bijection between two things that are not sets. And if they were sets the only invariant is cardinality. THe correct notion is categorical equivalence; I will edit my question accordingly. Nov 15, 2013 at 0:14