# Changes

## Linear representation theory of symmetric group:S5

, 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 $\overline{\mathbb{Q}}$ or $\mathbb{C}$) || 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]] $S_n$ of degree $n$ || 4 $n = 5$ || [[Family version::linear representation theory of symmetric groups]]
|-
| [[projective general linear group of degree two]] over a [[finite field]] of size $q$ || $q = 5$, i.e., [[field:F5]] , so the group is $PGL(2,5)$ || [[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 || || $\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$ || 1 + 1 + 1 + 1 + 1 || sign representation|-| sign representation || 1 || 1 + 1 + 1 + 1 + 1 || || $\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$ || 5 || trivial representation|-| standard representation || 4 || 4 + 1 || || $\frac{5!}{5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ || 2 + 1 + 1 + 1|| product of standard and sign representation|-| product of standard and sign representation || 4 || 2 + 1 + 1 + 1 || || $\frac{5!}{5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ || 4 + 1 || standard representation|-| irreducible five-dimensional representation || 5 || 3 + 2 || || $\frac{5!}{4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ || 2 + 2 + 1 || other irreducible five-dimensional representation|-| irreducible five-dimensional representation || 5 || 2 + 2 + 1 || || $\frac{5!}{4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ || 3 + 2|| other irreducible five-dimensional representation|-| exterior square of standard representation || 6 || 3 + 1 + 1 || || $\frac{5!}{5 \cdot 2 \cdot 2 \cdot 1 \cdot 1}$ || 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 $q$) !! Degree of each representation ($q = 5$) !! Number of representations (general odd $q$) !! Number of representations ($q = 5$) !! Sum of squares of degrees (general odd $q$) !! Sum of squares of degrees ($q = 5$) !! Symmetric group name|-| Trivial || -- || $x \mapsto 1$|| 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 $\{ \pm 1 \}$ || 1 || 1 || 1 || 1 || 1 || 1 || sign|- | Nontrivial component of permutation representation of $PGL_2$ on the projective line over $\mathbb{F}_q$ || -- || -- || $q$ || 5 || 1 || 1 || $q^2$ || 25 || irreducible 5D|-| Tensor product of sign representation and nontrivial component of permutation representation on projective line || -- || -- || $q$ || 5 || 1 || 1 || $q^2$ || 25 || other irreducible 5D|-| Induced from one-dimensional representation of Borel subgroup || ? || ? || $q + 1$ || 6 || $(q-3)/2$ || 1 || $(q+1)^2(q-3)/2$ || 36 || exterior square of standard representation|-| Unclear || a nontrivial homomorphism $\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast$, with the property that $\varphi(x)^{q+1} = 1$ for all $x$, and $\varphi$ takes values other than $\pm 1$. Identify $\varphi$ and $\varphi^q$. || unclear || $q - 1$ || 4 || $(q-1)/2$ || 2 || $(q-1)^3/2$ || 32 || standard representation, product of standard and sign|-! Total || NA || NA || NA || NA || $q + 2$ || 7 || $q^3 - q$ || 120 || NA|} ==Character table== {{character table facts to check against}}<section begin="character table"/>{| class="sortable" border="1"! Representation/conjugacy class representative and size !! $()$ (size 1) !! $(1,2)$ (size 10) !! $(1,2)(3,4)$ (size 15) !! $(1,2,3)$ (size 20) !! $(1,2,3)(4,5)$ (size 20) !! $(1,2,3,4,5)$ (size 24) !! $(1,2,3,4)$ (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 !! $()$ (size 1) !! $(1,2)$ (size 10) !! $(1,2)(3,4)$ (size 15) !! $(1,2,3)$ (size 20) !! $(1,2,3)(4,5)$ (size 20) !! $(1,2,3,4,5)$ (size 24) !! $(1,2,3,4)$ (size 30)
|-
| trivial representation || 1 || 5 10 || $\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$ 15 || 1 + 1 + 1 + 1 + 1 20 || sign representation20 || 24|| 30
|-
| sign representation || 1 || 1 + 1 + 1 + 1 + 1 -10 || $\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$ 15 || 5 20 || trivial representation-20 || 24|| -30
|-
| standard representation || 4 1 || 5 || 4 + 1 (or is it 2 + 1 + 1 + 1?) 0 || $\frac{5!}{|| -5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ || 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 || $\frac{5!}{|| 5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ || 4 + 1 (or is it 2 + 1 + 1 + 1?) -6 || standard representation0
|-
| irreducible five-dimensional representation || 5 1 || 2 || 3 + 2 || $\frac{5!}{-4 || 4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ || 2 + 2 + 1 0 || other irreducible five-dimensional representation6
|-
| irreducible five-dimensional representation || 5 1 || -2 + 2 + 1 || $\frac{5!}{4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ || 3 + 2-4 || other irreducible five-dimensional representation4 || 0 || 6
|-
| exterior square of standard representation || 6 1 || 3 + 1 + 1 0 || $\frac{-5!}{5 \cdot 2 \cdot 2 \cdot 1 \cdot 1}$ || 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>