# Linear representation theory of alternating group:A6

From Groupprops

This article gives specific information, namely, linear representation theory, about a particular group, namely: alternating group:A6.

View linear representation theory of particular groups | View other specific information about alternating group:A6

## Summary

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

Degrees of irreducible representations over a splitting field (such as or ) | 1,5,5,8,8,9,10 grouped form: 1 (1 time), 5 (2 times), 8 (2 times), 9 (1 time), 10 (1 time) maximum: 10, lcm: 360, number: 7, sum of squares: 360 |

Ring generated by character values | |

Minimal splitting field, i.e., field of realization of all irreducible representations (characteristic zero) | Quadratic extension of Same as field generated by character values |

Orbits of irreducible representations under action of automorphism group | orbits of size 1 for representations of degree 1,9,10; orbits of size two for degree 5 and degree 8 representations (the degree 8 representations are interchanged under conjugation by an odd permutation; the degree 5 representations are interchaged by an automorphism that is outer for as well) |

Orbits of irreducible representations under action of Galois group | orbits of size 1 for representations of degree 1,5,5,9,10; orbit of size two for degree 8 representations (automorphism ) |

Minimal splitting field in prime characteristic | Case : prime field Case : quadratic extension of |

Smallest size splitting field | field:F11 |

Degrees of irreducible representations over the rational numbers | 1,5,5,9,10,16 |

## Family contexts

Family name | Parameter values | General discussion of linear representation theory of family |
---|---|---|

alternating group | 6 | linear representation theory of alternating groups |

projective special linear group of degree two over a finite field of size | , i.e., field:F9, i.e., field of nine elements, so the group is | linear representation theory of projective special linear group of degree two over a finite field |

## GAP implementation

The degrees of irreducible representations can be found using GAP's CharacterDegrees and AlternatingGroup functions:

gap> CharacterDegrees(AlternatingGroup(6)); [ [ 1, 1 ], [ 5, 2 ], [ 8, 2 ], [ 9, 1 ], [ 10, 1 ] ]

This means that there is 1 degree 1 irreducible, 2 degree 5 irreducibles, 2 degree 8 irreducibles, 1 degree 9 irreducible, and 1 degree 10 irreducible representation.

The characters of irreducible representations can be computed using GAP's CharacterTable function:

gap> Irr(CharacterTable(AlternatingGroup(6))); [ Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 5, 1, 2, -1, -1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 5, 1, -1, 2, -1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 8, 0, -1, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ), Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 8, 0, -1, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ), Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 9, 1, 0, 0, 1, -1, -1 ] ), Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 10, -2, 1, 1, 0, 0, 0 ] ) ]