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

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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
 
| [[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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==Family contexts==
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{| class="sortable" border="1"
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! Family name !! Parameter values !! General discussion of linear representation theory of family
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| [[alternating group]] || 5 || [[linear representation theory of alternating groups]]
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| [[projective general linear group of degree two]] over a [[finite field]] || [[field:F4]] || [[linear representation theory of projective general linear group of degree two over a finite field]]
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| [[projective special linear group of degree two]] over a [[finite field]] || [[field:F5]] || [[linear representation theory of projective special linear group of degree two over a finite field]]
 
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Revision as of 01:42, 20 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

Family contexts

Family name Parameter values General discussion of linear representation theory of family
alternating group 5 linear representation theory of alternating groups
projective general linear group of degree two over a finite field field:F4 linear representation theory of projective general linear group of degree two over a finite field
projective special linear group of degree two over a finite field field:F5 linear representation theory of projective special linear group of degree two over a finite field

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 ] ) ]