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Linear representation theory of symmetric group:S5

6,158 bytes added, 05:41, 16 January 2013
Family contexts
==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 size [[splitting field]] || [[field:F7]], i.e., the 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]]
|}
{| 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|-| 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|-| 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 || 5 10 || <math>\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}</math> 15 || 1 + 1 + 1 + 1 + 1 20 || sign representation20 || 24|| 30
|-
| sign representation || 1 || 1 + 1 + 1 + 1 + 1 -10 || <math>\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}</math> 15 || 5 20 || trivial representation-20 || 24|| -30
|-
| standard representation || 4 1 || 5 || 4 + 1 (or is it 2 + 1 + 1 + 1?) 0 || <math>\frac{5!}{|| -5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}</math> || 2 + 1 + 1 + 1 (or is it 4 + 1?)-6 || product of standard and sign representation0
|-
| product of standard and sign representation || 4 1 || -5 || 2 + 1 + 1 + 1 (or is it 4 + 1?) 0 || <math>\frac{5!}{|| 5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}</math> || 4 + 1 (or is it 2 + 1 + 1 + 1?) -6 || standard representation0
|-
| irreducible five-dimensional representation || 5 1 || 2 || 3 + 2 || <math>\frac{5!}{-4 || 4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}</math> || 2 + 2 + 1 0 || other irreducible five-dimensional representation6
|-
| irreducible five-dimensional representation || 5 1 || -2 + 2 + 1 || <math>\frac{5!}{4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}</math> || 3 + 2-4 || other irreducible five-dimensional representation4 || 0 || 6
|-
| exterior square of standard representation || 6 1 || 3 + 1 + 1 0 || <math>\frac{-5!}{5 \cdot 2 \cdot 2 \cdot 1 \cdot 1}</math> || 3 + 1 + 1 0 || 0 || 4 || the same representation, because the partition is self-conjugate.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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