# Linear representation theory of alternating group:A5

This article gives specific information, namely, linear representation theory, about a particular group, namely: alternating group:A5.

View linear representation theory of particular groups | View other specific information about alternating group:A5

## Summary

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

Degrees of irreducible representations over a splitting field (such as or ) | 1,3,3,4,5 maximum: 5, lcm: 60, number: 5, sum of squares: 60 |

Ring generated by character values | or |

Minimal splitting field, i.e., smallest field of realization for all irreducible representations | (not sure -- need to verify!) -- quadratic extension of field of rational numbersSame as field generated by character values |

Orbit structure under action of automorphism group | orbits of size 1 each of degree 1, 4, and 5 representations, orbit of size 2 of degree 3 representations (interchanged via an automorphism induced by conjugation by odd permutation) |

Orbit structure under action of Galois group over rationals | orbits of size 1 each of degree 1, 4, and 5 representations, orbit of size 2 of degree 3 representations (interchanged via the mapping ) |

Degrees of irreducible representations over the field of rational numbers | 1,4,5,6 |

## Family contexts

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

alternating group | 5 | linear representation theory of alternating groups |

projective general linear group of degree two over a finite field | field:F4 | linear representation theory of projective general linear group of degree two over a finite field |

projective special linear group of degree two over a finite field | field:F5 | linear representation theory of projective special linear group of degree two over a finite field |

COMPARE AND CONTRAST: View linear representation theory of groups of order 60 to compare and contrast the linear representation theory with other groups of order 60.

## Degrees of irreducible representations

### Interpretation as alternating group

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

The partitions of 5 that are self-conjugate give irreducible representations of symmetric group:S5 that split into two irreducible representations of half the dimension each over alternating group:A5. Conjugate pairs of non-self-conjugate partitions of 5 restrict to equivalent irreducible representations over the alternating group.

Name(s) of representation(s) at symmetric groups level | Partition or pair of partitions | Self-conjugate case or conjugate pair case | Degree(s) for representations of symmetric group | Hook-length formula for degree | Degree(s) of representations for alternating group |
---|---|---|---|---|---|

trivial, sign | 5, 1 + 1 + 1 + 1 + 1 | conjugate pair case | 1, 1 | 1 | |

standard, product of sign and standard | 4 + 1, 2 + 1 + 1 + 1 | conjugate pair case | 4, 4 | 4 | |

irreducible five-dimensional | 3 + 2, 2 + 2 + 1 | conjugate pair case | 5, 5 | 5 | |

exterior square of standard | 3 + 1 + 1 | self-conjugate | 6 | 3, 3 |

## 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

Representation/conjugacy class representative and size | (size 1) | (size 15) | (size 20) | (size 12) | (size 12) |
---|---|---|---|---|---|

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

restriction of standard | 4 | 0 | 1 | -1 | -1 |

irreducible five-dimensional | 5 | 1 | -1 | 0 | 0 |

one restriction of exterior square of standard | 3 | -1 | 0 | ||

other restriction of exterior square of standard | 3 | -1 | 0 |

Below are the size-degree-weighted characters, obtained by multiplying each character value by the size of the conjugacy class and then dividing by the degree of the representation. Note that size-degree-weighted characters are algebraic integers.

Representation/conjugacy class representative and size | (size 1) | (size 15) | (size 20) | (size 12) | (size 12) |
---|---|---|---|---|---|

trivial | 1 | 15 | 20 | 12 | 12 |

restriction of standard | 1 | 0 | 5 | -3 | -3 |

irreducible five-dimensional | 1 | 3 | -4 | 0 | 0 |

one restriction of exterior square of standard | 1 | -5 | 0 | ||

other restriction of exterior square of standard | 1 | -5 | 0 |

## GAP implementation

The character degrees can be computed using GAP's CharacterDegrees function:

gap> CharacterDegrees(AlternatingGroup(5)); [ [ 1, 1 ], [ 3, 2 ], [ 4, 1 ], [ 5, 1 ] ]

This means that there is 1 degree 1 irreducible representation, 2 degree 3 irreducible representations, 1 degree 4 irreducible representation, and 1 degree 5 irreducible representation.

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

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