# Linear representation theory of general linear group of degree two over a finite field

From Groupprops

This article gives specific information, namely, linear representation theory, about a family of groups, namely: general linear group of degree two. This article restricts attention to the case where the underlying ring is a finite field.

View linear representation theory of group families | View other specific information about general linear group of degree two | View other specific information about group families for rings of the type finite field

This article describes the linear representation theory of the general linear group of degree two over a finite field. The order (size) of the field is , and the characteristic prime is . is a power of .

See also the linear representation theories of: special linear group of degree two, projective general linear group of degree two, and projective special linear group of degree two.

## Summary

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

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

number of irreducible representations | , equal to the number of conjugacy classes. See number of irreducible representations equals number of conjugacy classes, element structure of general linear group of degree two over a finite field#Conjugacy class structure |

maximum degree of irreducible representation over a splitting field | |

lcm of degrees of irreducible representations over a splitting field | Case odd: ; case even: |

sum of squares of degrees of irreducible representations over a splitting field | , equal to the order of the group. See sum of squares of degrees of irreducible representations equals group order. |

## Particular cases

Group | Order of the group | Number of irreducible representations | Degrees of irreducible representations | Linear representation theory page | ||
---|---|---|---|---|---|---|

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

general linear group:GL(2,3) | 3 | 3 | 48 | 8 | 1,1,2,2,2,3,3,4 | linear representation theory of general linear group:GL(2,3) |

direct product of A5 and Z3 | 2 | 4 | 180 | 15 | 1,1,1,3,3,3,3,3,3,4,4,4,5,5,5 | |

general linear group:GL(2,5) | 5 | 5 | 480 | 24 | 1 (4 times), 4 (10 times), 5 (4 times), 6 (6 times) | linear representation theory of general linear group:GL(2,5) |

general linear group:GL(2,7) | 7 | 7 | 2016 | 48 | 1 (6 times), 6 (21 times), 7 (6 times), 8 (15 times) | linear representation theory of general linear group:GL(2,7) |

## Irreducible representations

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

One-dimensional, factor through the determinant map | a homomorphism | 1 | |||

Unclear | a homomorphism , up to the equivalence , excluding the cases where | unclear | |||

Tensor product of one-dimensional representation and the nontrivial component of permutation representation of on the projective line over | a homomorphism | where is the nontrivial component of permutation representation of on the projective line over | |||

Induced from one-dimensional representation of Borel subgroup | homomorphisms with , where is treated as unordered. | Induced from the following representation of the Borel subgroup: | |||

Total | NA | NA | NA |