# Linear representation theory of symmetric group:S6

This article gives specific information, namely, linear representation theory, about a particular group, namely: symmetric group:S6.

View linear representation theory of particular groups | View other specific information about symmetric group:S6

This article describes the linear representation theory of symmetric group:S6, a group of order . We take this to be the group of permutations on the set .

## Summary

Item | Summary |
---|---|

Degrees of irreducible representations over a splitting field | 1,1,5,5,5,5,9,9,10,10,16 maximum: 16, lcm: 720, number: 11, sum of squares: 720 |

Schur index values of irreducible representations | 1,1,1,1,1,1,1,1,1,1,1 |

Smallest ring of realization for all irreducible representations (characteristic zero) | -- ring of integers |

Smallest field of realization for all irreducible representations, i.e., smallest splitting field (characteristic zero) | -- hence it is a rational representation group |

Criterion for a field to be a splitting field | Any field of characteristic not 2,3, or 5 |

Smallest size splitting field | field:F7 |

## Family contexts

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

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

## Degrees of irreducible representations

FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroupSize bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroupCumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization

Note that the linear representation theory of the symmetric group of degree six works over any field of characteristic not equal to 2, 3, or 5, and the list of degrees is .

### Interpretation as symmetric group

Common name of representation | Degree | Partition corresponding to representation | Hook length formula for degree | Conjugate partition | Representation for conjugate partition |
---|---|---|---|---|---|

trivial representation | 1 | 6 | 1 + 1 + 1 + 1 + 1 + 1 | sign representation | |

sign representation | 1 | 1 + 1 + 1 + 1 + 1 + 1 | 6 | trivial representation | |

standard representation | 5 | 5 + 1 | 2 + 1 + 1 + 1 + 1 | product of standard and sign representation | |

product of standard and sign representation | 5 | 2 + 1 + 1 + 1 + 1 | 5 + 1 | standard representation | |

irreducible five-dimensional representation | 5 | 3 + 3 | 2 + 2 + 2 | other irreducible five-dimensional representation | |

irreducible five-dimensional representation | 5 | 2 + 2 + 2 | 3 + 3 | other irreducible five-dimensional representation | |

irreducible nine-dimensional representation | 9 | 4 + 2 | 2 + 2 + 1 + 1 | other irreducible nine-dimensional representation | |

irreducible nine-dimensional representation | 9 | 2 + 2 + 1 + 1 | 2 + 2 + 1 + 1 | other irreducible nine-dimensional representation | |

second exterior power of the standard representation | 10 | 4 + 1 + 1 | 3 + 1 + 1 + 1 | other irreducible ten-dimensional representation | |

third exterior power of the standard representation | 10 | 3 + 1 + 1 + 1 | 4 + 1 + 1 | other irreducible ten-dimensional representation | |

irreducible sixteen-dimensional representation | 16 | 3 + 2 + 1 | 3 + 2 + 1 | same, i.e., self-conjugate |

## 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

The character table below is incomplete, it contains only eight of the eleven representations.

Representation/conjugacy class representative and size | (size 1) | (size 15) | (size 15) | (size 40) | (size 40) | (size 45) | (size 90) | (size 90) | (size 120) | (size 120) | -- size 144 |
---|---|---|---|---|---|---|---|---|---|---|---|

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

sign | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 |

standard | 5 | 3 | -1 | 2 | -1 | 1 | 1 | -1 | 0 | -1 | 0 |

product of standard and sign | 5 | -3 | 1 | 2 | -1 | 1 | -1 | -1 | 0 | 1 | 0 |

other five-dimensional irreducible | 5 | -1 | 3 | -1 | 2 | 1 | 1 | -1 | -1 | 0 | 0 |

other five-dimensional irreducible | 5 | 1 | -3 | -1 | 2 | 1 | -1 | -1 | 1 | 0 | 0 |

second exterior power of standard | 10 | 2 | -2 | 1 | 1 | -2 | 0 | 0 | -1 | 1 | 0 |

third exterior power of standard | 10 | -2 | 2 | 1 | 1 | -2 | 0 | 0 | 1 | -1 | 0 |

## GAP implementation

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

gap> CharacterDegrees(SymmetricGroup(6)); [ [ 1, 2 ], [ 5, 4 ], [ 9, 2 ], [ 10, 2 ], [ 16, 1 ] ]

This means there are 2 degree 1 irreducible representations, 4 degree 5 irreducible representations, 2 degree 9 irreducible representations, 2 degree 10 irreducible representations, and 1 degree 16 irreducible representation.

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

gap> Irr(CharacterTable(SymmetricGroup(6))); [ Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 5, -3, 1, 1, 2, 0, -1, -1, -1, 0, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 9, -3, 1, -3, 0, 0, 0, 1, 1, -1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 5, -1, 1, 3, -1, -1, 2, 1, -1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 10, -2, -2, 2, 1, 1, 1, 0, 0, 0, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 16, 0, 0, 0, -2, 0, -2, 0, 0, 1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 5, 1, 1, -3, -1, 1, 2, -1, -1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 10, 2, -2, -2, 1, -1, 1, 0, 0, 0, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 9, 3, 1, 3, 0, 0, 0, -1, 1, -1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 5, 3, 1, -1, 2, 0, -1, 1, -1, 0, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ) ]