Linear representation theory of symmetric group:S7 - Revision history

Vipul: /* Summary */

2012-04-30T00:46:54Z

Summary

← Older revision		Revision as of 00:46, 30 April 2012
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	==Summary==		==Summary==

			<section begin="summary"/>
	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Item !! Value		! Item !! Value
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	\| smallest size splitting field \|\| [[field:F11]], i.e., the field with 11 elements		\| smallest size splitting field \|\| [[field:F11]], i.e., the field with 11 elements
	\|}		\|}
			<section end="summary"/>

	==Family contexts==		==Family contexts==

Vipul at 16:40, 22 May 2011

2011-05-22T16:40:53Z

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	{{group-specific information\|		{{group-specific information\|
	group = symmetric group:S7\|		group = symmetric group:S7\|
	information type = linear representation theory}}		information type = linear representation theory\|
			connective = of}}

	==Summary==		==Summary==

Vipul at 15:59, 22 May 2011

2011-05-22T15:59:24Z

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	\|-		\|-
	\| [[Schur index]] values of irreducible representations over a splitting field \|\| 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 (all 1s)		\| [[Schur index]] values of irreducible representations over a splitting field \|\| 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 (all 1s)
	\|-
	~~\| condition for a field to be a splitting field \|\| any field of characteristic not equal to 2,3,5,7~~
	\|-		\|-
	\| smallest ring of realization (characteristic zero) \|\| <math>\mathbb{Z}</math> -- [[ring of integers]]		\| smallest ring of realization (characteristic zero) \|\| <math>\mathbb{Z}</math> -- [[ring of integers]]

Vipul at 15:41, 22 May 2011

2011-05-22T15:41:19Z

← Older revision		Revision as of 15:41, 22 May 2011
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	\|-		\|-
	\| smallest field of realization (characteristic zero) \|\| <math>\mathbb{Q}</math> -- [[field of rational numbers]]		\| smallest field of realization (characteristic zero) \|\| <math>\mathbb{Q}</math> -- [[field of rational numbers]]
			\|-
			\| condition for a field to be a splitting field \|\| Any field of characteristic not 2, 3, 5, or 7.
	\|-		\|-
	\| smallest size splitting field \|\| [[field:F11]], i.e., the field with 11 elements		\| smallest size splitting field \|\| [[field:F11]], i.e., the field with 11 elements

Vipul: /* Summary */

2011-05-10T19:02:39Z

Summary

← Older revision		Revision as of 19:02, 10 May 2011
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	! Item !! Value		! Item !! Value
	\|-		\|-
	\| [[degrees of irreducible representations]] over a [[splitting field]] \|\| 1,1,6,6,14,14,14,14,15,15,20,21,21,35,35<br>[[maximum degree of irreducible representation\|maximum]]: 35, [[lcm of degrees of irreducible representations\|lcm]]: 420, [[number of irreducible representations\|number]]: 15, [[sum of squares of degrees of irreducible representations equals group order\|sum of squares]]: 5040		\| [[degrees of irreducible representations]] over a [[splitting field]] \|\| 1,1,6,6,14,14,14,14,15,15,20,21,21,35,35<br>[[maximum degree of irreducible representation\|maximum]]: 35, [[lcm of degrees of irreducible representations\|lcm]]: 420, [[number of irreducible representations equals number of conjugacy classes\|number]]: 15, [[sum of squares of degrees of irreducible representations equals group order\|sum of squares]]: 5040
	\|-		\|-
	\| [[Schur index]] values of irreducible representations over a splitting field \|\| 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 (all 1s)		\| [[Schur index]] values of irreducible representations over a splitting field \|\| 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 (all 1s)

Vipul: Created page with "{{group-specific information| group = symmetric group:S7| information type = linear representation theory}} ==Summary== {| class="sortable" border="1" ! Item !! Value |- | [[de..."

2011-05-10T19:02:16Z

Created page with "{{group-specific information| group = symmetric group:S7| information type = linear representation theory}} ==Summary== {| class="sortable" border="1" ! Item !! Value |- | [[de..."

New page

{{group-specific information|
group = symmetric group:S7|
information type = linear representation theory}}

==Summary==

{| class="sortable" border="1"
! Item !! Value
|-
| [[degrees of irreducible representations]] over a [[splitting field]] || 1,1,6,6,14,14,14,14,15,15,20,21,21,35,35<br>[[maximum degree of irreducible representation|maximum]]: 35, [[lcm of degrees of irreducible representations|lcm]]: 420, [[number of irreducible representations|number]]: 15, [[sum of squares of degrees of irreducible representations equals group order|sum of squares]]: 5040
|-
| [[Schur index]] values of irreducible representations over a splitting field || 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 (all 1s)
|-
| condition for a field to be a splitting field || any field of characteristic not equal to 2,3,5,7
|-
| smallest ring of realization (characteristic zero) || <math>\mathbb{Z}</math> -- [[ring of integers]]
|-
| smallest field of realization (characteristic zero) || <math>\mathbb{Q}</math> -- [[field of rational numbers]]
|-
| smallest size splitting field || [[field:F11]], i.e., the field with 11 elements
|}

==Family contexts==

{| class="sortable" border="1"
! Family name !! Parameter values !! General discussion of linear representation theory of family
|-
| [[symmetric group]] || 7 || [[linear representation theory of symmetric groups]]
|}

==GAP implementation==

The [[degrees of irreducible representations]] can be computed using GAP's [[GAP:CharacterDegrees|CharacterDegrees]] and [[GAP:SymmetricGroup|SymmetricGroup]] functions:

<pre>gap> CharacterDegrees(SymmetricGroup(7));
[ [ 1, 2 ], [ 6, 2 ], [ 14, 4 ], [ 15, 2 ], [ 20, 1 ], [ 21, 2 ], [ 35, 2 ] ]</pre>

The charaters of irreducible representations can be computed using GAP's [[GAP:CharacterTable|CharacterTable]] function:

<pre>gap> Irr(CharacterTable(SymmetricGroup(7)));
[ Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, -1, 1 ] ), Character( CharacterTable( Sym(
[ 1 .. 7 ] ) ), [ 6, -4, 2, 0, 3, -1, -1, 0, -2, 0, 1, 1, 1, 0, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ),
[ 14, -6, 2, -2, 2, 0, 2, -1, 0, 0, 0, -1, -1, 1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 14, -4, 2, 0, -1, -1, -1, 2, 2, 0, -1, -1,
1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 15, -5, -1, 3, 3, 1, -1, 0, -1, -1, -1, 0, 0, 0, 1 ] ),
Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 35, -5, -1, -1, -1, 1, -1, -1, 1, 1, 1, 0, 0, -1, 0 ] ), Character( CharacterTable( Sym(
[ 1 .. 7 ] ) ), [ 21, -1, 1, 3, -3, -1, 1, 0, 1, -1, 1, 1, -1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ),
[ 21, 1, 1, -3, -3, 1, 1, 0, -1, -1, -1, 1, 1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 20, 0, -4, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0,
-1 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 35, 5, -1, 1, -1, -1, -1, -1, -1, 1, -1, 0, 0, 1, 0 ] ), Character( CharacterTable( Sym(
[ 1 .. 7 ] ) ), [ 14, 4, 2, 0, -1, 1, -1, 2, -2, 0, 1, -1, -1, 0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ),
[ 15, 5, -1, -3, 3, -1, -1, 0, 1, -1, 1, 0, 0, 0, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 14, 6, 2, 2, 2, 0, 2, -1, 0, 0, 0, -1, 1,
-1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 7 ] ) ), [ 6, 4, 2, 0, 3, 1, -1, 0, 2, 0, -1, 1, -1, 0, -1 ] ), Character( CharacterTable( Sym(
[ 1 .. 7 ] ) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ) ]</pre>