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

 $(\sum\lambda_{i}g_{i})\circ v=\sum\lambda_{i}(g_{i}\cdot v).$

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

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

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

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

 $\underline{\overline{\mathbb{V}}}=\mathbb{V};$
 $\overline{\underline{M}}=M.$

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

Title representations vs modules RepresentationsVsModules 2013-03-22 19:18:59 2013-03-22 19:18:59 joking (16130) joking (16130) 4 joking (16130) Definition msc 20C99