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

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| [[field generated by character values]] (characteristic zero) || <math>\mathbb{Q}(\sqrt{-2})</math>, same as <math>\mathbb{Q}[t]/(t^2 + 2)</math> | | [[field generated by character values]] (characteristic zero) || <math>\mathbb{Q}(\sqrt{-2})</math>, same as <math>\mathbb{Q}[t]/(t^2 + 2)</math> | ||

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## Revision as of 00:22, 2 September 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 |

## 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 |

## GAP implementation

### Degrees of irreducible representations

The degrees of irreducible representations can be computed using GAP's CharacterDegrees function, as follows:

gap> CharacterDegrees(GL(2,3)); [ [ 1, 2 ], [ 2, 3 ], [ 3, 2 ], [ 4, 1 ] ]

### Character table

The character table can be computed using GAP's CharacterTable function, as follows:

gap> Irr(CharacterTable(GL(2,3))); [ Character( CharacterTable( GL(2,3) ), [ 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( GL(2,3) ), [ 1, 1, 1, 1, 1, -1, -1, -1 ] ), Character( CharacterTable( GL(2,3) ), [ 2, -1, 2, -1, 2, 0, 0, 0 ] ), Character( CharacterTable( GL(2,3) ), [ 2, 1, -2, -1, 0, -E(8)-E(8)^3, E(8)+E(8)^3, 0 ] ), Character( CharacterTable( GL(2,3) ), [ 2, 1, -2, -1, 0, E(8)+E(8)^3, -E(8)-E(8)^3, 0 ] ), Character( CharacterTable( GL(2,3) ), [ 3, 0, 3, 0, -1, 1, 1, -1 ] ), Character( CharacterTable( GL(2,3) ), [ 3, 0, 3, 0, -1, -1, -1, 1 ] ), Character( CharacterTable( GL(2,3) ), [ 4, -1, -4, 1, 0, 0, 0, 0 ] ) ]

A visual display of the character table can be achieved as follows:

gap> Display(CharacterTable(GL(2,3))); CT1 2 4 1 4 1 3 3 3 2 3 1 1 1 1 . . . . 1a 6a 2a 3a 4a 8a 8b 2b X.1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 -1 -1 -1 X.3 2 -1 2 -1 2 . . . X.4 2 1 -2 -1 . A -A . X.5 2 1 -2 -1 . -A A . X.6 3 . 3 . -1 1 1 -1 X.7 3 . 3 . -1 -1 -1 1 X.8 4 -1 -4 1 . . . . A = -E(8)-E(8)^3 = -Sqrt(-2) = -i2

### Irreducible representations

The irreducible representations of can be computed using GAP's GAP:IrreducibleRepresentations function, as follows:

gap> IrreducibleRepresentations(GL(2,3)); [ CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ -1 ] ], [ [ 1 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ 0, E(3) ], [ E(3)^2, 0 ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ -1/2*E(24)^11-1/2*E(24)^17, -1/2*E(24)-E(24)^11-E(24)^17-1/2*E(24)^19 ], [ -1/2*E(8)-1/2*E(8)^3, 1/2*E(24)^11+1/2*E(24)^17 ] ], [ [ E(3), E(3) ], [ 0, E(3)^2 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ E(24)+E(24)^19, -1 ], [ -E(3)+E(3)^2, -E(24)-E(24)^19 ] ], [ [ E(3)+2*E(3)^2, E(8)+E(8)^3 ], [ -E(24)-E(24)^19, -E(3)^2 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ 0, 0, 1 ], [ 0, -1, 0 ], [ 1, 0, 0 ] ], [ [ 1, 0, 0 ], [ 0, 0, 1 ], [ 1, -1, -1 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ], [ [ 1, 0, 0 ], [ -1, -1, -1 ], [ 0, 1, 0 ] ] ], <action isomorphism> ), CompositionMapping( [ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ] -> [ [ [ 0, 0, -E(3), 0 ], [ 0, 0, -E(3), -E(3) ], [ -E(3)^2, 0, 0, 0 ], [ E(3)^2, -E(3)^2, 0, 0 ] ], [ [ -E(3)^2, E(3)^2, 0, 0 ], [ -E(3)^2, 0, 0, 0 ], [ 0, 0, E(3)^2, 0 ], [ 0, 0, 1, 1 ] ] ], <action isomorphism> ) ]