Linear representation theory of alternating group:A8

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    ] ) ]