# Linear representation theory of Klein four-group

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This article gives specific information, namely, linear representation theory, about a particular group, namely: Klein four-group.

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The Klein four-group is an example of a rational representation group in the sense that all its representations can be realized over the field of rational numbers. In fact, all its representations can be realized over the two-element set and are one-dimensional, hence the representations described for characteristic zero generalize to any situation where the characteristic is not two.

We describe the Klein four-group as a four-element group with identity element and three non-identity elements . Recall that each of the non-identity elements has order two and the product of any two distinct ones among them is the third one.

## Representations

There are four irreducible representations, all one-dimensional:

- The trivial representation: This sends all four elements to .
- The representation with kernel : This sends and to and sends and to .
- The representation with kernel : This sends and to and sends and to .
- The representation with kernel : This sends and to and sends and to .

## Character table

Rep/element | ||||
---|---|---|---|---|

trivial | 1 | 1 | 1 | 1 |

kernel | 1 | 1 | -1 | -1 |

kernel | 1 | -1 | 1 | -1 |

kernel | 1 | -1 | -1 | 1 |

## Realizability information

### Smallest ring of realization

Representation | Smallest ring over which it is realized |
Smallest set of elements in matrix entries |
---|---|---|

trivial representation | -- ring of integers | |

any of the nontrivial representations | -- ring of integers |

### Smallest ring of realization as orthogonal matrices

Representation | Smallest ring over which it is realized |
Smallest set of elements in matrix entries |
---|---|---|

trivial representation | -- ring of integers | |

any of the nontrivial representations | -- ring of integers |