# 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

## Contents

## Summary

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

Degrees of irreducible representations over a splitting field (such as or ) | 1,3,3,4,5 grouped form: 1 (1 time), 3 (2 times), 4 (1 time), 5 (1 time) maximum: 5, lcm: 60, number: 5, sum of squares: 60, quasirandom degree: 3 |

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

Ring generated by character values | or |

Minimal splitting field, i.e., smallest field of realization for all irreducible representations | -- quadratic extension of field of rational numbers Same 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 of size | , i.e., field:F4, so the group is | 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 of size | , i.e., field:F5, so the group is | 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 |

## Back story

This page gives information about the degrees of irreducible representations, character table, and irreducible linear representations of alternating group:A5. It does not, however, provide an adequate explanation of how one might arrive at (i.e.,deduce) the information. For more on the back story, see determination of character table of alternating group:A5.

## 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 irreducible constituent of restriction of exterior square of standard | 3 | -1 | 0 | ||

other irreducible constituent of 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

### Degrees of irreducible representations

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.

### Character table

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 ] ) ]

The character table can be displayed somewhat more nicely as follows:

gap> Display(CharacterTable(AlternatingGroup(5))); CT2 2 2 2 . . . 3 1 . 1 . . 5 1 . . 1 1 1a 2a 3a 5a 5b 2P 1a 1a 3a 5b 5a 3P 1a 2a 1a 5b 5a 5P 1a 2a 3a 1a 1a X.1 1 1 1 1 1 X.2 3 -1 . A *A X.3 3 -1 . *A A X.4 4 . 1 -1 -1 X.5 5 1 -1 . . A = -E(5)-E(5)^4 = (1-ER(5))/2 = -b5

### Irreducible representations

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

gap> IrreducibleRepresentations(AlternatingGroup(5)); [ [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ], [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ 0, -1, 0 ], [ -E(5)-E(5)^4, -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4, 1 ], [ 2*E(5)+E(5)^2+E(5)^3+2*E(5)^4, 2*E(5)+E(5)^2+E(5)^3+2*E(5)^4, -1 ] ], [ [ 1, -E(5)-E(5)^4, -E(5)-E(5)^4 ], [ 0, -1, -1 ], [ 0, 1, 0 ] ] ] , [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ 1, 0, 1 ], [ -1, 0, 0 ], [ -1, -1, E(5)+E(5\ )^4 ] ], [ [ 0, -E(5)^2-E(5)^3, E(5)^2+E(5)^3 ], [ -1, -1, E(5)+E(5)^4 ], [ -E(5)-E(5)^4, -E(5)-E(5)^4, 1 ] ] ], [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ E(3)^2, -1/3*E(3)-2/3*E(3)^2, E(3)^2, 4/3*E(3)+2/3*E(3)^2 ], [ E(3), 4/3*E(3)+2/3*E(3)^2, 1, -1/3*E(3)-2/3*E(3)^2 ], [ E(3), E(3), 0, -E(3)^2 ], [ E(3), -2/3*E(3)-1/3*E(3)^2, E(3), -1/3*E(3)-2/3*E(3)^2 ] ], [ [ E(3), 1/3*E(3)-1/3*E(3)^2, 0, 2/3*E(3)-2/3*E(3)^2 ], [ E(3)^2, 0, 0, 0 ], [ 0, 2/3*E(3)+1/3*E(3)^2, 1, -2/3*E(3)-1/3*E(3)^2 ], [ 1, 1, 0, -E(3) ] ] ], [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ -1, -1, -1, -1, -1 ], [ 1, 1, 0, 0, 1 ], [ 0, -1, -1, 0, -1 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 1, 1 ] ], [ [ -1, -1, -1, -1, -1 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1 ], [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ] ] ] ]