Difference between revisions of "Linear representation theory of alternating group:A5"

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(Created page with "{{group-specific information| group = alternating group:A5| information type = linear representation theory| connective = of}} ==Summary== {| class="sortable" border="1" ! Item...")
 
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| [[degrees of irreducible representations]] over a [[splitting field]] || 1,3,4,4,5<br>[[maximum degree of irreducible representation|maximum]]: 5, [[lcm of degrees of irreducible representations|lcm]]: 60, [[number of irreducible representations equals number of conjugacy classes|number]]: 5
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| [[degrees of irreducible representations]] over a [[splitting field]] || 1,3,3,4,5<br>[[maximum degree of irreducible representation|maximum]]: 5, [[lcm of degrees of irreducible representations|lcm]]: 60, [[number of irreducible representations equals number of conjugacy classes|number]]: 5
 
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|}
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==GAP implementation==
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The character degrees can be computed using GAP's [[GAP:CharacterDegrees|CharacterDegrees]] function:
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<pre>gap> CharacterDegrees(AlternatingGroup(5));
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[ [ 1, 1 ], [ 3, 2 ], [ 4, 1 ], [ 5, 1 ] ]</pre>
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This means that there is 1 degree 1 irreducible representation, 2 degree 3 irreducible representations, 1 degree 4 irreducible representation, and 1 degree 5 irreducible representation.
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The characters of irreducible representations can be computed using GAP's [[GAP:CharacterTable|CharacterTable]] function:
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<pre>gap> Irr(CharacterTable(AlternatingGroup(5)));
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[ Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 1, 1, 1, 1, 1 ] ),
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  Character( CharacterTable( Alt( [ 1 .. 5 ] ) ),
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    [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ), Character( CharacterTable( Alt(
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    [ 1 .. 5 ] ) ), [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ),
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  Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 4, 0, 1, -1, -1 ] ),
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  Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 5, 1, -1, 0, 0 ] ) ]</pre>

Revision as of 17:12, 10 May 2011

This article gives specific information, namely, linear representation theory, about a particular group, namely: alternating group:A5.
View linear representation theory of particular groups | View other specific information about alternating group:A5

Summary

Item Value
degrees of irreducible representations over a splitting field 1,3,3,4,5
maximum: 5, lcm: 60, number: 5

GAP implementation

The character degrees can be computed using GAP's CharacterDegrees function:

gap> CharacterDegrees(AlternatingGroup(5));
[ [ 1, 1 ], [ 3, 2 ], [ 4, 1 ], [ 5, 1 ] ]

This means that there is 1 degree 1 irreducible representation, 2 degree 3 irreducible representations, 1 degree 4 irreducible representation, and 1 degree 5 irreducible representation.

The characters of irreducible representations can be computed using GAP's CharacterTable function:

gap> Irr(CharacterTable(AlternatingGroup(5)));
[ Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 1, 1, 1, 1, 1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 5 ] ) ),
    [ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ), Character( CharacterTable( Alt(
    [ 1 .. 5 ] ) ), [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ),
  Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 4, 0, 1, -1, -1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 5, 1, -1, 0, 0 ] ) ]