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

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

View linear representation theory of group families | View other specific information about projective general linear group of degree two

This article describes the linear representation theory of the projective 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 . The group is denoted or .

See also the linear representation theory for: special linear group, projective special linear group, and general linear group.

## Summary

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

degrees of irreducible representations over a splitting field | Case odd: 1 (2 times), ( times), (2 times), ( times) Case even: 1 (1 time), ( times), (1 time), ( times) |

number of irreducible representations | Case odd: , case even: See number of irreducible representations equals number of conjugacy classes, element structure of projective general linear group of degree two over a finite field#Conjugacy class structure |

quasirandom degree (minimum degree of nontrivial ireducible representation) | 1 |

maximum degree of irreducible representation | |

lcm of degrees of irreducible representations | Case odd: ; Case even: |

sum of squares of degrees of irreducible representations | , equal to the group order; see sum of squares of degrees of irreducible representations equals group order |

## Particular cases

## Irreducible representations

### Case , odd

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 |
---|---|---|---|---|---|

Trivial | -- | 1 | 1 | 1 | |

Sign representation | -- | Kernel is projective special linear group of degree two, image is | 1 | 1 | 1 |

Unclear | a nontrivial homomorphism , with the property that for all , and takes values other than . Identify and . | unclear | |||

Nontrivial component of permutation representation of on the projective line over | -- | -- | 1 | ||

Tensor product of sign representation and nontrivial component of permutation representation on projective line | -- | -- | 1 | ||

Induced from one-dimensional representation of Borel subgroup | homomorphism , with taking values other than , up to inverses. | Induced from the following representation of the image of the Borel subgroup: | |||

Total | NA | NA | NA |