# Linear representation theory of general affine group of degree one over a finite field

From Groupprops

This article gives specific information, namely, linear representation theory, about a family of groups, namely: general affine group of degree one.

View linear representation theory of group families | View other specific information about general affine group of degree one

This article discusses the linear representation theory of the general affine group of degree one over a finite field of size and characteristic , where .

## Summary

Item | Value |
---|---|

degrees of irreducible representations over a splitting field | 1 ( times), (1 time) |

number of irreducible representations | See number of irreducible representations equals number of conjugacy classes, element structure of general affine group of degree one over a finite field |

maximum degree of irreducible representation over a splitting field | |

lcm of degrees of irreducible representations over a splitting field | |

sum of squares of degrees of irreducible representations | sum of squares of degrees of irreducible representations equals group order |

condition for a field to be a splitting field | The field should be a splitting field for the multiplicative group and its characteristic should not be . Equivalently, its order should be relatively prime to and and it should contain all the primitive roots of unity, i.e., the polynomial must split over the field. |

smallest size splitting field | The finite field whose size is the smallest prime power other than that is congruent to 1 modulo . |

## Particular cases

(field size) | (underlying prime, field characteristic) | Group | Order of the group | Second part of GAP ID | Degrees of irreducible representations | Number of irreducible representations | Linear representation theory page |
---|---|---|---|---|---|---|---|

2 | 2 | cyclic group:Z2 | 2 | 1 | 1,1 | 2 | linear representation theory of cyclic group:Z2 |

3 | 3 | symmetric group:S3 | 6 | 1 | 1,1,2 | 3 | linear representation theory of symmetric group:S3 |

4 | 2 | alternating group:A4 | 12 | 3 | 1,1,1,3 | 4 | linear representation theory of alternating group:A4 |

5 | 5 | general affine group:GA(1,5) | 20 | 3 | 1,1,1,1,4 | 5 | linear representation theory of general affine group:GA(1,5) |

7 | 7 | general affine group:GA(1,7) | 42 | 1 | 1,1,1,1,1,1,6 | 7 | linear representation theory of general affine group:GA(1,7) |

8 | 2 | general affine group:GA(1,8) | 56 | 11 | 1,1,1,1,1,1,1,7 | 8 | linear representation theory of general affine group:GA(1,8) |

9 | 3 | general affine group:GA(1,9) | 72 | 39 | 1,1,1,1,1,1,1,1,8 | 9 | linear representation theory of general affine group:GA(1,9) |

## Irreducible representations

We denote by the pair the element . The multiplicative part is and the additive part is .

Description of collection of representations | Parameter for describing the representation | How the representation is described | Degree of each representation | Number of representations | Sum of squares of degrees |
---|---|---|---|---|---|

One-dimensional, depends only on multiplicative part, kernel contains additive subgroup | a homomorphism | 1 | |||

Nontrivial component of permutation representation on elements. | None | Consider the permutation representation of induced by its action on . This has a one-dimensional trivial subrepresentation. The complementary subrepresentation has degree and is irreducible. | 1 | ||

Total | -- | -- | -- |