Linear representation theory of alternating group:A6

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

gap> CharacterDegrees(AlternatingGroup(6)); [ [ 1, 1 ], [ 5, 2 ], [ 8, 2 ], [ 9, 1 ], [ 10, 1 ] ]

This means that there is 1 degree 1 irreducible, 2 degree 5 irreducibles, 2 degree 8 irreducibles, 1 degree 9 irreducible, and 1 degree 10 irreducible representation.

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

gap> Irr(CharacterTable(AlternatingGroup(6))); [ Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 5, 1, 2, -1, -1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 5, 1, -1, 2, -1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 8, 0, -1, -1, 0, -E(5)-E(5)^4,     -E(5)^2-E(5)^3 ] ), Character( CharacterTable( Alt( [ 1 .. 6 ] ) ),    [ 8, 0, -1, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ), Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 9, 1, 0, 0, 1, -1, -1 ] ), Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 10, -2, 1, 1, 0, 0, 0 ] ) ]