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 $n$ || $n = 5$ || [[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 $q$ || $q = 5$, i.e., [[field:F5]] , so the group is $PGL(2,5)$ || [[Family version::linear representation theory of projective general linear group of degree twoover a finite field]]
|}
| 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 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)
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| trivial representation || 1 || 1 || 1 || 1 || 1 || 1 || 1
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| sign representation || 1 || -1 || 1 || 1 || -1 || 1 || -1
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| standard representation || 4 || 2 || 0 || 1 || -1 || -1|| 0
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| product of standard and sign representation || 4 || -2 || 0 || 1 || 1 || -1|| 0
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| irreducible five-dimensional representation || 5 || 1 || 1 || -1 || 1 || 0 || -1
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| irreducible five-dimensional representation || 5 || -1 || 1 || -1 || -1 || 0|| 1
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| 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) !! $(1,2,3)(4,45)$ (size 3020) !! $(1,2,3)(,4,5)$ (size 2024) !! $(1,2,3,4,5)$ (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
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| standard representation || 4 1 || 2 5 || 0 || 1 5 || 0 -5 || -1 6 || -10
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| product of standard and sign representation || 4 1 || -2 5 || 0 || 1 5 || 0 5 || 1 -6 || -10
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| irreducible five-dimensional representation || 5 1 || ? 2 || ? 3 || ? -4 || ? 4 || ? 0 || ?-6
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| 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
|}