Linear representation theory of alternating group:A8

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This article gives specific information, namely, linear representation theory, about a particular group, namely: alternating group:A8.
View linear representation theory of particular groups | View other specific information about alternating group:A8

Summary

Item Value
degrees of irreducible representations over a splitting field 1,7,14,20,21,21,21,28,35,45,45,56,64,70
grouped form: 1 (1 time), 7 (1 time) 14 (1 time), 20 (1 time), 21 (3 times), 28 (1 time), 35 (1 time), 45 (2 times), 56 (1 time), 64 (1 time), 70 (1 time)
maximum: 70, lcm: 20160, number: 14, sum of squares: 20160
minimal splitting field, i.e., smallest field of realization of irreducible representations (characteristic zero) \mathbb{Q}(\omega + \omega^2 + \omega^4, \zeta + \zeta^2 + \zeta^4 + \zeta^8) where \omega is a primitive seventh root of unity and \zeta is a primitive fifteenth root of unity
Same as \mathbb{Q}(\sqrt{-7},\sqrt{-15})
condition for a field of characteristic not 2,3,5,7 to be a splitting field Both -7 and -15 should be squares in the field
minimal splitting field, i.e., smallest field of realization of irreducible representations, prime characteristic not equal to 2,3,5,7 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]


Family contexts

Family name Parameter values General discussion of linear representation theory of family
alternating group 8 linear representation theory of alternating groups
projective special linear group degree (order of matrices) 4, field:F2 linear representation theory of projective special linear groups

GAP implementation

The degrees of irreducible representations can be computed using GAP's CharacterDegrees and AlternatingGroup functions:

gap> CharacterDegrees(AlternatingGroup(8));
[ [ 1, 1 ], [ 7, 1 ], [ 14, 1 ], [ 20, 1 ], [ 21, 3 ], [ 28, 1 ], [ 35, 1 ], [ 45, 2 ], [ 56, 1 ], [ 64, 1 ], [ 70, 1 ] ]

This means that there is 1 degree 1 irreducible 1 degree 7 irreducible, 1 degree 14 irreducible, 1 degree 20 irreducible, 3 degree 21 irreducibles, and so on.

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

gap> Irr(CharacterTable(AlternatingGroup(8)));
[ Character( CharacterTable( Alt( [ 1 .. 8 ] ) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( Alt( [ 1 .. 8 ] ) ),
    [ 7, 3, -1, 4, 0, 1, 1, -1, 2, -1, -1, -1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 8 ] ) ), [ 14, 2, 6, -1, -1, 2, 0, 2, -1, -1, -1, 0, 0, 0
     ] ), Character( CharacterTable( Alt( [ 1 .. 8 ] ) ), [ 20, 4, 4, 5, 1, -1, 0, 0, 0, 0, 0, 1, -1, -1 ] ), Character( CharacterTable( Alt(
    [ 1 .. 8 ] ) ), [ 21, 1, -3, 6, -2, 0, -1, 1, 1, 1, 1, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 8 ] ) ), [ 21, 1, -3, -3, 1, 0, -1, 1, 1,
      -E(15)-E(15)^2-E(15)^4-E(15)^8, -E(15)^7-E(15)^11-E(15)^13-E(15)^14, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 8 ] ) ),
    [ 21, 1, -3, -3, 1, 0, -1, 1, 1, -E(15)^7-E(15)^11-E(15)^13-E(15)^14, -E(15)-E(15)^2-E(15)^4-E(15)^8, 0, 0, 0 ] ), Character( CharacterTable( Alt(
    [ 1 .. 8 ] ) ), [ 28, 4, -4, 1, 1, 1, 0, 0, -2, 1, 1, -1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 8 ] ) ), [ 35, -5, 3, 5, 1, 2, -1, -1, 0,
      0, 0, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 8 ] ) ), [ 45, -3, -3, 0, 0, 0, 1, 1, 0, 0, 0, 0, E(7)^3+E(7)^5+E(7)^6, E(7)+E(7)^2+E(7)^4
     ] ), Character( CharacterTable( Alt( [ 1 .. 8 ] ) ), [ 45, -3, -3, 0, 0, 0, 1, 1, 0, 0, 0, 0, E(7)+E(7)^2+E(7)^4, E(7)^3+E(7)^5+E(7)^6 ] ),
  Character( CharacterTable( Alt( [ 1 .. 8 ] ) ), [ 56, 0, 8, -4, 0, -1, 0, 0, 1, 1, 1, -1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 8 ] ) ),
    [ 64, 0, 0, 4, 0, -2, 0, 0, -1, -1, -1, 0, 1, 1 ] ), Character( CharacterTable( Alt( [ 1 .. 8 ] ) ), [ 70, 2, -2, -5, -1, 1, 0, -2, 0, 0, 0, 1, 0, 0
     ] ) ]