Linear representation theory of alternating group:A10

Degrees of irreducible representations
These can be computed using the CharacterDegrees function:

gap> CharacterDegrees(CharacterTable(AlternatingGroup(10))); [ [ 1, 1 ], [ 9, 1 ], [ 35, 1 ], [ 36, 1 ], [ 42, 1 ], [ 75, 1 ], [ 84, 1 ], [ 90, 1 ], [ 126, 1 ], [ 160, 1 ], [ 210, 1 ], [ 224, 2 ], [ 225, 1 ], [ 252, 1 ], [ 288, 1 ], [ 300, 1 ], [ 315, 1 ], [ 350, 1 ], [ 384, 2 ], [ 450, 1 ], [ 525, 1 ], [ 567, 1 ] ]

Character table
These can be computed using the Irr and CharacterTable functions:

gap> Irr(CharacterTable(AlternatingGroup(10))); [ Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 9, 5, 1, 6, 2, 3, -1, 0, 3, -1, 0, 1, 4, 0, 1, -1, 1, -1, 2, -1, -1, -1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 35, 11, 3, 14, 2, 2, 2, -1, 3, 3, 0, -1, 5, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, -1 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 36, 8, -4, 15, -1, 3, -1, 0, 2, -2, -1, 0, 6, -2, 0, 1, -1, 1, 1, 1, 1, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 42, 6, 2, 0, 0, 3, 3, -3, 0, -4, 0, 2, -3, 1, 0, 2, -1, -1, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 75, 15, 3, 15, 3, 0, 0, 3, 1, -3, 1, -1, 0, 0, 0, 0, 0, 0, -2, 1, 1, -1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 84, 0, -4, 21, -3, 3, 3, 3, -2, 2, 1, 0, 4, 0, 1, -1, -1, -1, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 90, 14, 2, 6, 2, 3, -1, 0, 0, 4, 0, 2, -5, -1, 1, 0, -1, 1, -1, -1, -1, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 126, -14, 6, 21, 1, 6, -2, 0, -4, 0, -1, -2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 160, 16, 0, 34, -2, -2, -2, -2, 0, 0, 0, 0, 5, 1, -1, 0, 0, 0, -1, -1, -1, 0, 1, 1 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 210, 6, -6, -21, 3, 0, 0, 3, -4, 0, -1, 2, 5, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 224, -16, 0, 14, 2, 2, 2, -1, 0, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, -1, 2 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 224, -16, 0, 14, 2, 2, 2, -1, 0, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 2, -1 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 225, 5, 9, 15, -1, -6, 2, 0, -1, 3, -1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, -1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 252, 8, 4, -21, -1, 3, -1, 0, -2, 2, 1, 0, 2, -2, -1, 2, 1, -1, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 288, 16, 0, -6, -2, 6, -2, 0, 0, 0, 0, 0, -7, 1, -1, -2, 0, 0, 1, 1, 1, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 300, 0, 4, -15, -3, 3, 3, 3, 2, -2, -1, 0, 0, 0, 0, 0, 1, 1, -1, -1, -1, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 315, 19, -5, 21, 1, -3, 1, 0, -1, -1, -1, -1, -5, -1, 1, 0, 1, -1, 0, 0, 0, 1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 350, -10, -2, 35, -1, -1, -1, -1, -2, -2, 1, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, -1 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 384, 0, 0, -24, 0, 0, 0, -3, 0, 0, 0, 0, 4, 0, 1, -1, 0, 0, -1,     E(21)^2+E(21)^8+E(21)^10+E(21)^11+E(21)^13+E(21)^19, E(21)+E(21)^4+E(21)^5+E(21)^16+E(21)^17+E(21)^20, 0, 0, 0 ] ), Character( CharacterTable( Alt(    [ 1 .. 10 ] ) ), [ 384, 0, 0, -24, 0, 0, 0, -3, 0, 0, 0, 0, 4, 0, 1, -1, 0, 0, -1, E(21)+E(21)^4+E(21)^5+E(21)^16+E(21)^17+E(21)^20,      E(21)^2+E(21)^8+E(21)^10+E(21)^11+E(21)^13+E(21)^19, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 450, 10, 2, -15, 1, -3, 1, 0,      -2, -2, 1, -2, 0, 0, 0, 0, -1, 1, 2, -1, -1, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 525, -15, 5, 0, 0, -3, -3, 3, 3, -1,      0, 1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 1, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 10 ] ) ), [ 567, -9, -9, 0, 0, 0, 0, 0, 3, 3, 0, -1, -3, 1,      0, 2, 0, 0, 0, 0, 0, -1, 0, 0 ] ) ]