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Linear representation theory of symmetric group:S5

1,572 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> || <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 group of degree twoover a finite field]]
|}
| 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 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)(4,45)</math> (size 3020) !! <math>(1,2,3)(,4,5)</math> (size 2024) !! <math>(1,2,3,4,5)</math> (size 2430)
|-
| trivial representation || 1 || 1 10 || 1 15 || 1 20 || 1 20 || 1 24|| 130
|-
| sign representation || 1 || -1 10 || 1 15 || 1 20 || -1 20 || -1 24|| 1-30
|-
| standard representation || 4 1 || 2 5 || 0 || 1 5 || 0 -5 || -1 6 || -10
|-
| product of standard and sign representation || 4 1 || -2 5 || 0 || 1 5 || 0 5 || 1 -6 || -10
|-
| irreducible five-dimensional representation || 5 1 || ? 2 || ? 3 || ? -4 || ? 4 || ? 0 || ?-6
|-
| irreducible five-dimensional representation || 5 1 || ? -2 || ? 3 || ? -4 || ? -4 || ? 0 || ?6
|-
| exterior square of standard representation || 6 1 || 0 || -2 5 || 0 || 0 || 0 4 || 10
|}
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