# Difference between revisions of "Linear representation theory of symmetric group:S5"

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information type = linear representation theory| | information type = linear representation theory| | ||

connective = of}} | connective = of}} | ||

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+ | This article describes the linear representation theory of [[symmetric group:S5]], a group of order <math>120</math>. We take this to be the group of permutations on the set <math>\{1,2,3,4,5 \}</math>. | ||

==Summary== | ==Summary== | ||

+ | <section begin="summary"/> | ||

{| class="sortable" border="1" | {| class="sortable" border="1" | ||

! Item !! Value | ! Item !! Value | ||

|- | |- | ||

− | | [[Degrees of irreducible representations]] over a splitting field || 1,1,4,4,5,5,6 | + | | [[Degrees of irreducible representations]] over a [[splitting field]] (such as <math>\overline{\mathbb{Q}}</math> or <math>\mathbb{C}</math>) || 1,1,4,4,5,5,6<br>[[maximum degree of irreducible representation|maximum]]: 6, [[lcm of degrees of irreducible representations|lcm]]: 60, [[number of irreducible representations equals number of conjugacy classes|number]]: 7, [[sum of squares of degrees of irreducible representations equals order of group|sum of squares]]: 120 |

+ | |- | ||

+ | | [[Schur index]] values of irreducible representations || 1,1,1,1,1,1,1<br>[[maximum Schur index of irreducible representation|maximum]]: 1, [[lcm of Schur indices of irreducible representations|lcm]]: 1 | ||

|- | |- | ||

| Smallest ring of realization for all irreducible representations (characteristic zero) || <math>\mathbb{Z}</math> -- ring of integers | | Smallest ring of realization for all irreducible representations (characteristic zero) || <math>\mathbb{Z}</math> -- ring of integers | ||

|- | |- | ||

− | | Smallest field of realization for all irreducible representations (characteristic zero) || <math>\mathbb{Q}</math> -- hence it is a [[rational representation group]] | + | | Smallest field of realization for all irreducible representations, i.e., smallest splitting field (characteristic zero) || <math>\mathbb{Q}</math> -- hence it is a [[rational representation group]] |

+ | |- | ||

+ | | Criterion for a field to be a [[splitting field]] || Any field of characteristic not equal to 2,3, or 5. | ||

|- | |- | ||

− | | | + | | Smallest size [[splitting field]] || [[field:F7]], i.e., the field of 7 elements. |

|} | |} | ||

− | + | <section end="summary"/> | |

==Family contexts== | ==Family contexts== | ||

Line 23: | Line 30: | ||

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

|- | |- | ||

− | | [[symmetric group]] || | + | | [[symmetric group]] <math>S_n</math> of degree <math>n</math> || <math>n = 5</math> || [[Family version::linear representation theory of symmetric groups]] |

|- | |- | ||

− | | [[projective general linear group of degree two]] || [[field:F5]] || [[linear representation theory of projective general linear | + | | [[projective general linear group of degree two]] over a [[finite field]] of size <math>q</math> || <math>q = 5</math>, i.e., [[field:F5]], so the group is <math>PGL(2,5)</math> || [[Family version::linear representation theory of projective general linear group of degree two over a finite field]] |

|} | |} | ||

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Note that the linear representation theory of the symmetric group of degree four works over any field of characteristic not equal to two or three, and the list of degrees is <math>1,1,4,4,5,5,6</math>. | Note that the linear representation theory of the symmetric group of degree four works over any field of characteristic not equal to two or three, and the list of degrees is <math>1,1,4,4,5,5,6</math>. | ||

− | === | + | ===Interpretation as symmetric group=== |

{| class="sortable" border="1" | {| class="sortable" border="1" | ||

− | ! Common name of representation !! Degree !! | + | ! Common name of representation !! Degree !! Corresponding [[unordered integer partition|partition]] !! Young diagram !! [[Hook-length formula]] for degree !! Conjugate partition !! Representation for conjugate partition |

|- | |- | ||

− | | trivial representation || 1 || 5 || <math>\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}</math> || 1 + 1 + 1 + 1 + 1 || sign representation | + | | trivial representation || 1 || 5 || || <math>\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}</math> || 1 + 1 + 1 + 1 + 1 || sign representation |

+ | |- | ||

+ | | sign representation || 1 || 1 + 1 + 1 + 1 + 1 || || <math>\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}</math> || 5 || trivial representation | ||

+ | |- | ||

+ | | standard representation || 4 || 4 + 1 || || <math>\frac{5!}{5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}</math> || 2 + 1 + 1 + 1|| product of standard and sign representation | ||

+ | |- | ||

+ | | product of standard and sign representation || 4 || 2 + 1 + 1 + 1 || || <math>\frac{5!}{5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}</math> || 4 + 1 || standard representation | ||

+ | |- | ||

+ | | irreducible five-dimensional representation || 5 || 3 + 2 || || <math>\frac{5!}{4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}</math> || 2 + 2 + 1 || other irreducible five-dimensional representation | ||

+ | |- | ||

+ | | irreducible five-dimensional representation || 5 || 2 + 2 + 1 || || <math>\frac{5!}{4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}</math> || 3 + 2|| other irreducible five-dimensional representation | ||

+ | |- | ||

+ | | exterior square of standard representation || 6 || 3 + 1 + 1 || || <math>\frac{5!}{5 \cdot 2 \cdot 2 \cdot 1 \cdot 1}</math> || 3 + 1 + 1 || the same representation, because the partition is self-conjugate. | ||

+ | |} | ||

+ | |||

+ | ===Interpretation as projective general linear group of degree two=== | ||

+ | |||

+ | {{quotation|Compare and contrast with [[linear representation theory of projective general linear group of degree two over a finite field]]}} | ||

+ | |||

+ | Below is an interpretation of the group as the [[projective general linear group of degree two]] over [[field:F5]], the field of five elements. | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Description of collection of representations !! Parameter for describing each representation !! How the representation is described !! Degree of each representation (general odd <math>q</math>) !! Degree of each representation (<math>q = 5</math>) !! Number of representations (general odd <math>q</math>) !! Number of representations (<math>q = 5</math>) !! Sum of squares of degrees (general odd <math>q</math>) !! Sum of squares of degrees (<math>q = 5</math>) !! Symmetric group name | ||

|- | |- | ||

− | | | + | | Trivial || -- || <math>x \mapsto 1</math>|| 1 || 1 || 1 || 1 || 1 || 1 || trivial |

|- | |- | ||

− | | | + | | Sign representation || -- || Kernel is [[projective special linear group of degree two]] (in this case, [[alternating group:A5]]), image is <math>\{ \pm 1 \}</math> || 1 || 1 || 1 || 1 || 1 || 1 || sign |

+ | |- | ||

+ | | Nontrivial component of permutation representation of <math>PGL_2</math> on the projective line over <math>\mathbb{F}_q</math> || -- || -- || <math>q</math> || 5 || 1 || 1 || <math>q^2</math> || 25 || irreducible 5D | ||

|- | |- | ||

− | | product of | + | | Tensor product of sign representation and nontrivial component of permutation representation on projective line || -- || -- || <math>q</math> || 5 || 1 || 1 || <math>q^2</math> || 25 || other irreducible 5D |

|- | |- | ||

− | | | + | | Induced from one-dimensional representation of Borel subgroup || ? || ? || <math>q + 1</math> || 6 || <math>(q-3)/2</math> || 1 || <math>(q+1)^2(q-3)/2</math> || 36 || exterior square of standard representation |

|- | |- | ||

− | | | + | | Unclear || a nontrivial homomorphism <math>\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast</math>, with the property that <math>\varphi(x)^{q+1} = 1</math> for all <math>x</math>, and <math>\varphi</math> takes values other than <math>\pm 1</math>. Identify <math>\varphi</math> and <math>\varphi^q</math>. || unclear || <math>q - 1</math> || 4 || <math>(q-1)/2</math> || 2 || <math>(q-1)^3/2</math> || 32 || standard representation, product of standard and sign |

|- | |- | ||

− | | | + | ! Total || NA || NA || NA || NA || <math>q + 2</math> || 7 || <math>q^3 - q</math> || 120 || NA |

|} | |} | ||

+ | |||

+ | ==Character table== | ||

+ | |||

+ | {{character table facts to check against}} | ||

+ | <section begin="character table"/> | ||

+ | {| class="sortable" border="1" | ||

+ | ! Representation/conjugacy class representative and size !! <math>()</math> (size 1) !! <math>(1,2)</math> (size 10) !! <math>(1,2)(3,4)</math> (size 15) !! <math>(1,2,3)</math> (size 20) !! <math>(1,2,3)(4,5)</math> (size 20) !! <math>(1,2,3,4,5)</math> (size 24) !! <math>(1,2,3,4)</math> (size 30) | ||

+ | |- | ||

+ | | trivial representation || 1 || 1 || 1 || 1 || 1 || 1 || 1 | ||

+ | |- | ||

+ | | sign representation || 1 || -1 || 1 || 1 || -1 || 1 || -1 | ||

+ | |- | ||

+ | | standard representation || 4 || 2 || 0 || 1 || -1 || -1|| 0 | ||

+ | |- | ||

+ | | product of standard and sign representation || 4 || -2 || 0 || 1 || 1 || -1|| 0 | ||

+ | |- | ||

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

+ | |- | ||

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

+ | |- | ||

+ | | exterior square of standard representation || 6 || 0 || -2 || 0 || 0 || 1|| 0 | ||

+ | |} | ||

+ | <section end="character table"/> | ||

+ | Below are the size-degree-weighted characters, i.e., these are obtained by multiplying the 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]]. | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Representation/conjugacy class representative and size !! <math>()</math> (size 1) !! <math>(1,2)</math> (size 10) !! <math>(1,2)(3,4)</math> (size 15) !! <math>(1,2,3)</math> (size 20) !! <math>(1,2,3)(4,5)</math> (size 20) !! <math>(1,2,3,4,5)</math> (size 24) !! <math>(1,2,3,4)</math> (size 30) | ||

+ | |- | ||

+ | | trivial representation || 1 || 10 || 15 || 20 || 20 || 24|| 30 | ||

+ | |- | ||

+ | | sign representation || 1 || -10 || 15 || 20 || -20 || 24|| -30 | ||

+ | |- | ||

+ | | standard representation || 1 || 5 || 0 || 5 || -5 || -6 || 0 | ||

+ | |- | ||

+ | | product of standard and sign representation || 1 || -5 || 0 || 5 || 5 || -6 || 0 | ||

+ | |- | ||

+ | | irreducible five-dimensional representation || 1 || 2 || 3 || -4 || 4 || 0 || -6 | ||

+ | |- | ||

+ | | irreducible five-dimensional representation || 1 || -2 || 3 || -4 || -4 || 0 || 6 | ||

+ | |- | ||

+ | | exterior square of standard representation || 1 || 0 || -5 || 0 || 0 || 4 || 0 | ||

+ | |} | ||

+ | |||

+ | ==GAP implementation== | ||

+ | |||

+ | The degrees of irreducible representations can be computed using GAP's [[GAP:CharacterDegrees|CharacterDegrees]] function: | ||

+ | |||

+ | <pre>gap> CharacterDegrees(SymmetricGroup(5)); | ||

+ | [ [ 1, 2 ], [ 4, 2 ], [ 5, 2 ], [ 6, 1 ] ]</pre> | ||

+ | |||

+ | This means that there are 2 degree 1 irreducible representations, 2 degree 4 irreducible representations, 2 degree 5 irreducible representations, and 1 degree 6 irreducible representation. | ||

+ | |||

+ | The characters of all irreducible representations can be computed in full using GAP's [[GAP:CharacterTable|CharacterTable]] function: | ||

+ | |||

+ | <pre>gap> Irr(CharacterTable(SymmetricGroup(5))); | ||

+ | [ Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 1, -1, 1, 1, -1, -1, 1 ] ), | ||

+ | Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 4, -2, 0, 1, 1, 0, -1 ] ), | ||

+ | Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 5, -1, 1, -1, -1, 1, 0 ] ), | ||

+ | Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 6, 0, -2, 0, 0, 0, 1 ] ), | ||

+ | Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 5, 1, 1, -1, 1, -1, 0 ] ), | ||

+ | Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 4, 2, 0, 1, -1, 0, -1 ] ), | ||

+ | Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 1, 1, 1, 1, 1, 1, 1 ] ) ]</pre> |

## Latest revision as of 05:41, 16 January 2013

This article gives specific information, namely, linear representation theory, about a particular group, namely: symmetric group:S5.

View linear representation theory of particular groups | View other specific information about symmetric group:S5

This article describes the linear representation theory of symmetric group:S5, a group of order . We take this to be the group of permutations on the set .

## Summary

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

Degrees of irreducible representations over a splitting field (such as or ) | 1,1,4,4,5,5,6 maximum: 6, lcm: 60, number: 7, sum of squares: 120 |

Schur index values of irreducible representations | 1,1,1,1,1,1,1 maximum: 1, lcm: 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 equal to 2,3, or 5. |

Smallest size splitting field | field:F7, i.e., the field of 7 elements. |

## Family contexts

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

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

projective general linear group of degree two over a finite field of size | , i.e., field:F5, so the group is | linear representation theory of projective general linear group of degree two over a finite field |

## 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 four works over any field of characteristic not equal to two or three, and the list of degrees is .

### Interpretation as symmetric group

Common name of representation | Degree | Corresponding partition | Young diagram | Hook-length formula for degree | Conjugate partition | Representation for conjugate partition |
---|---|---|---|---|---|---|

trivial representation | 1 | 5 | 1 + 1 + 1 + 1 + 1 | sign representation | ||

sign representation | 1 | 1 + 1 + 1 + 1 + 1 | 5 | trivial representation | ||

standard representation | 4 | 4 + 1 | 2 + 1 + 1 + 1 | product of standard and sign representation | ||

product of standard and sign representation | 4 | 2 + 1 + 1 + 1 | 4 + 1 | standard representation | ||

irreducible five-dimensional representation | 5 | 3 + 2 | 2 + 2 + 1 | other irreducible five-dimensional representation | ||

irreducible five-dimensional representation | 5 | 2 + 2 + 1 | 3 + 2 | other irreducible five-dimensional representation | ||

exterior square of standard representation | 6 | 3 + 1 + 1 | 3 + 1 + 1 | the same representation, because the partition is self-conjugate. |

### Interpretation as projective general linear group of degree two

Compare and contrast with linear representation theory of projective general linear group of degree two over a finite field

Below is an interpretation of the group as the projective general linear group of degree two over field:F5, the field of five elements.

Description of collection of representations | Parameter for describing each representation | How the representation is described | Degree of each representation (general odd ) | Degree of each representation () | Number of representations (general odd ) | Number of representations () | Sum of squares of degrees (general odd ) | Sum of squares of degrees () | Symmetric group name |
---|---|---|---|---|---|---|---|---|---|

Trivial | -- | 1 | 1 | 1 | 1 | 1 | 1 | trivial | |

Sign representation | -- | Kernel is projective special linear group of degree two (in this case, alternating group:A5), image is | 1 | 1 | 1 | 1 | 1 | 1 | sign |

Nontrivial component of permutation representation of on the projective line over | -- | -- | 5 | 1 | 1 | 25 | irreducible 5D | ||

Tensor product of sign representation and nontrivial component of permutation representation on projective line | -- | -- | 5 | 1 | 1 | 25 | other irreducible 5D | ||

Induced from one-dimensional representation of Borel subgroup | ? | ? | 6 | 1 | 36 | exterior square of standard representation | |||

Unclear | a nontrivial homomorphism , with the property that for all , and takes values other than . Identify and . | unclear | 4 | 2 | 32 | standard representation, product of standard and sign | |||

Total | NA | NA | NA | NA | 7 | 120 | NA |

## 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 10) | (size 15) | (size 20) | (size 20) | (size 24) | (size 30) |
---|---|---|---|---|---|---|---|

trivial representation | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

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

standard representation | 4 | 2 | 0 | 1 | -1 | -1 | 0 |

product of standard and sign representation | 4 | -2 | 0 | 1 | 1 | -1 | 0 |

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

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

exterior square of standard representation | 6 | 0 | -2 | 0 | 0 | 1 | 0 |

Below are the size-degree-weighted characters, i.e., these are obtained by multiplying the 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 10) | (size 15) | (size 20) | (size 20) | (size 24) | (size 30) |
---|---|---|---|---|---|---|---|

trivial representation | 1 | 10 | 15 | 20 | 20 | 24 | 30 |

sign representation | 1 | -10 | 15 | 20 | -20 | 24 | -30 |

standard representation | 1 | 5 | 0 | 5 | -5 | -6 | 0 |

product of standard and sign representation | 1 | -5 | 0 | 5 | 5 | -6 | 0 |

irreducible five-dimensional representation | 1 | 2 | 3 | -4 | 4 | 0 | -6 |

irreducible five-dimensional representation | 1 | -2 | 3 | -4 | -4 | 0 | 6 |

exterior square of standard representation | 1 | 0 | -5 | 0 | 0 | 4 | 0 |

## GAP implementation

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

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

This means that there are 2 degree 1 irreducible representations, 2 degree 4 irreducible representations, 2 degree 5 irreducible representations, and 1 degree 6 irreducible representation.

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

gap> Irr(CharacterTable(SymmetricGroup(5))); [ Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 1, -1, 1, 1, -1, -1, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 4, -2, 0, 1, 1, 0, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 5, -1, 1, -1, -1, 1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 6, 0, -2, 0, 0, 0, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 5, 1, 1, -1, 1, -1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 4, 2, 0, 1, -1, 0, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 1, 1, 1, 1, 1, 1, 1 ] ) ]