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

Question

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 $F[G]$

Answer 1

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.