# Difference between revisions of "Linear representation theory of general linear group:GL(2,3)"

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| three-dimensional, factors through tensor product of standard and sign representations of <math>S_4</math> || 3 || 3 || -1 || 1 || 1 || 0 || 0 || -1 | | three-dimensional, factors through tensor product of standard and sign representations of <math>S_4</math> || 3 || 3 || -1 || 1 || 1 || 0 || 0 || -1 | ||

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− | | four-dimensional, induced from one-dimensional representation of Borel subgroup || 4 || -4 || 0 || 0 || 0 || | + | | four-dimensional, induced from one-dimensional representation of Borel subgroup || 4 || -4 || 0 || 0 || 0 || 1 || -1 || 0 |

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## Revision as of 18:19, 31 August 2013

This article gives specific information, namely, linear representation theory, about a particular group, namely: general linear group:GL(2,3).

View linear representation theory of particular groups | View other specific information about general linear group:GL(2,3)

This article describes the linear representation theory (in characteristic zero and other characteristics excluding 2,3) of general linear group:GL(2,3), which is the general linear group of degree two over field:F3.

## Summary

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

degrees of irreducible representations over a splitting field | 1,1,2,2,2,3,3,4 maximum: 4, lcm: 12, number: 8, sum of squares: 48 |

ring generated by character values (characteristic zero) | , same as |

field generated by character values (characteristic zero) | , same as |

other groups having the same character table | binary octahedral group, see linear representation theory of binary octahedral group. |

## Irreducible representations

### Interpretation as general linear group of degree two

The group is a general linear group of degree two over field:F3. Compare with linear representation theory of general linear group of degree two over a finite field.

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

One-dimensional, factor through the determinant map | a homomorphism | 1 | 1 | 2 | 2 | |||

Unclear | a homomorphism | unclear | 2 | 3 | 12 | |||

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 | 3 | 2 | 18 | |||

Induced from one-dimensional representation of Borel subgroup | Both distinct representations homomorphisms | Induced from the following representation of the Borel subgroup: | 4 | 1 | 16 | |||

Total | NA | NA | NA | NA | 8 | 48 |

## Character table

FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field):Orthogonality relations: Character orthogonality theorem | Column orthogonality theoremSeparation results(basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions|Character determines representation in characteristic zeroNumerical facts: Characters are cyclotomic integers | Size-degree-weighted characters are algebraic integersCharacter value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class| Conjugacy class of more than average size has character value zero for some irreducible character | Zero-or-scalar lemma

In the table below, we denote by a fixed square root of -2.

Representation/conjugacy class representative and size | (size 1) | (size 1) | (size 6) | (size 6) | (size 6) | (size 8) | (size 8) | (size 12) |
---|---|---|---|---|---|---|---|---|

trivial | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

nontrivial one-dimensional | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 |

two-dimensional (unclear) | 2 | 2 | 2 | 0 | 0 | -1 | -1 | 0 |

two-dimensional (unclear) | 2 | -2 | 0 | -1 | 1 | 0 | ||

two-dimensional (unclear) | 2 | -2 | 0 | -1 | 1 | 0 | ||

three-dimensional, factors through standard representation of symmetric group:S4 | 3 | 3 | -1 | -1 | -1 | 0 | 0 | 1 |

three-dimensional, factors through tensor product of standard and sign representations of | 3 | 3 | -1 | 1 | 1 | 0 | 0 | -1 |

four-dimensional, induced from one-dimensional representation of Borel subgroup | 4 | -4 | 0 | 0 | 0 | 1 | -1 | 0 |