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I am trying to understand the equivalence between group representations, $(V, \rho)$, and left modules over the group ring $F[G]$. Can you explain explicitly why it is the same?

My progress: Consider a group $G$. If $(V, \rho)$ is a representation of $G$, we can take $V$ to be a left module over $F[G]$ by defining: $gv = \rho(g)v$. So given a representation we can get a left module over $F[G]$.

The other direction is more confusing. Given a left module $M$ over $F[G]$, what is the vector space? Is $M$ necessarily a vector space? (The concept of a module is rather new to me).

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    A vector space over $F$ is the same thing as an $F$-module (exercise!). An $F[G]$-module must, in particular, be an $F$-module, so it's an $F$-vector space.2011-06-16
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    Check also that an $F[G]$-module map is the same thing as a $G$-equivariant $F$-linear map between the representation spaces.2011-06-16
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    Ok. I got it. Thank you both.2011-06-16
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    It would be great if you posted what you have as an answer. Then people could look at it and leave comments.2011-06-16

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