representations vs modules

Let $G$ be a group and $k$ a field. Recall that a pair $(V,\cdot )$ is a representation of $G$ over $k$, if $V$ is a vector space over $k$ and $\cdot :G\times V\to V$ is a linear group action (compare with parent object). On the other hand we have a group algebra $kG$, which is a vector space over $k$ with $G$ as a basis and the multiplication is induced from the multiplication in $G$. Thus we can consider modules over $kG$. These two concepts are related.

If $𝕍=(V,\cdot )$ is a representation of $G$ over $k$, then define a $kG$-module $\overline{𝕍}$ by putting $\overline{𝕍}=V$ as a vector space over $k$ and the action of $kG$ on $\overline{𝕍}$ is given by

$$(\sum {\lambda }_i{g}_i)\circ v=\sum {\lambda }_i(g_i\cdot v).$$

It can be easily checked that $\overline{𝕍}$ is indeed a $kG$-module.

Analogously if $M$ is a $kG$-module (with action denoted by ,,$\circ$”), then the pair $\underset{¯}{M}=(M,\cdot )$ is a representation of $G$ over $k$, where ,,$\cdot$” is given by

$$g\cdot v=g\circ v.$$

As a simple exercise we leave the following proposition to the reader:

Proposition. Let $𝕍$ be a representation of $G$ over $k$ and let $M$ be a $kG$-module. Then

$$\overline{\underset{¯}{𝕍}}=𝕍;$$

$$\overline{\underset{¯}{M}}=M.$$

This means that modules and representations are the same concept. One can generalize this even further by showing that $\overline{\cdot }$ and $\underset{¯}{\cdot }$ are both functors, which are (mutualy invert) isomorphisms of appropriate categories.

Therefore we can easily define such concepts as ,,direct sum of representations” or ,,tensor product of representations”, etc.