# 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 $\underset{\xaf}{M}=(M,\cdot )$ 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 $\mathbb{V}$ be a representation of $G$ over $k$ and let $M$ be a $kG$-module. Then

$$\overline{\underset{\xaf}{\mathbb{V}}}=\mathbb{V};$$ |

$$\overline{\underset{\xaf}{M}}=M.$$ |

This means that modules and representations are the same concept. One can generalize this even further by showing that $\overline{\cdot}$ and $\underset{\xaf}{\cdot}$ 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.

Title | representations vs modules |
---|---|

Canonical name | RepresentationsVsModules |

Date of creation | 2013-03-22 19:18:59 |

Last modified on | 2013-03-22 19:18:59 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 20C99 |