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Linear representation theory of symmetric group:S5

2,504 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]]
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| [[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== {{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
|}
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