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Linear representation theory of symmetric group:S5

6,859 bytes added, 05:41, 16 January 2013
Family contexts
information type = linear representation theory|
connective = of}}
 
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==
<section begin="summary"/>
{| class="sortable" border="1"
! Item !! Value
|-
| [[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 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.
|-
| Criterion for a field to be a Smallest size [[splitting field ]] || Any [[field of characteristic not equal to 2:F7]],3i.e., or 5the field of 7 elements.
|}
<section end="summary"/>
==Family contexts==
! Family name !! Parameter values !! General discussion of linear representation theory of family
|-
| [[symmetric group]] <math>S_n</math> of degree <math>n</math> || 4 <math>n = 5</math> || [[Family version::linear representation theory of symmetric groups]]
|-
| [[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 groups group of degree twoover a finite field]]
|}
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>.
===Explanation of degrees from the perspective of Interpretation as symmetric group of degree five===
{| class="sortable" border="1"
! Common name of representation !! Degree !! Partition corresponding to representation 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|-| 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
|-
| sign representation Trivial || 1 || 1 + 1 + 1 + 1 + 1 -- || <math>x \frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot mapsto 1}</math> || 5 1 || 1 || 1 || 1 || 1 || 1 || trivial representation
|-
| standard Sign representation || 4 -- || 4 + 1 Kernel is [[projective special linear group of degree two]] (or in this case, [[alternating group:A5]]), image is it 2 + 1 + 1 + 1?) || <math>\frac{5!}{5 \cdot 3 \cdot 2 \cdot pm 1 \cdot 1}</math> || 2 + 1 + || 1 || 1 || 1 + || 1 (or is it 4 + || 1?)|| product sign|- | Nontrivial component of standard and sign permutation representationof <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
|-
| Tensor product of standard sign representation and sign nontrivial component of permutation representation on projective line || 4 -- || -- || <math>q</math> || 5 || 2 + 1 + 1 + 1 (or is it 4 + || 1?) || <math>\frac{5!}{5 \cdot 3 \cdot q^2 \cdot 1 \cdot 1}</math> || 4 + 1 (or is it 2 + 1 + 1 + 1?) 25 || standard representationother irreducible 5D
|-
| irreducible fiveInduced from one-dimensional representation of Borel subgroup || 5 ? || 3 ? || <math>q + 2 1</math> || 6 || <math>\frac{5!}{4 \cdot (q-3 \cdot )/2 \cdot </math> || 1 \cdot || <math>(q+1})^2(q-3)/2</math> || 2 + 2 + 1 36 || other irreducible five-dimensional exterior square of standard representation
|-
| irreducible five-dimensional representation || 5 || 2 + 2 + 1 Unclear || a nontrivial homomorphism <math>\fracvarphi:\mathbb{5!F}_{4 q^2}^\ast \to \cdot 3 mathbb{C}^\cdot 2 ast</math>, with the property that <math>\cdot varphi(x)^{q+1} = 1 </math> for all <math>x</math>, and <math>\cdot 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> || other irreducible five-dimensional 32 || standard representation, product of standard and sign
|-
! Total || NA | exterior square of standard representation |NA | 6 |NA | 3 + 1 + 1 | NA || <math>\frac{5!}{5 \cdot 2 \cdot q + 2 \cdot 1 \cdot 1}</math> || 7 || <math>q^3 + 1 + 1 - q</math> || 120 || the same representation, because the partition is self-conjugate.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>
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