Linear representation theory of alternating group:A5

Interpretation as alternating group
The partitions of 5 that are self-conjugate give irreducible representations of symmetric group:S5 that split into two irreducible representations of half the dimension each over alternating group:A5. Conjugate pairs of non-self-conjugate partitions of 5 restrict to equivalent irreducible representations over the alternating group.

Character table




Below are the size-degree-weighted characters, obtained by multiplying each character value by the size of the conjugacy class and then dividing by the degree of the representation. Note that size-degree-weighted characters are algebraic integers.

Degrees of irreducible representations
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.

Character table
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 ] ) ]

The character table can be displayed somewhat more nicely as follows:

gap> Display(CharacterTable(AlternatingGroup(5))); CT2

2 2  2  .  .  .     3  1  .  1  .  .     5  1  .  .  1  1

1a 2a 3a 5a 5b 2P 1a 1a 3a 5b 5a 3P 1a 2a 1a 5b 5a 5P 1a 2a 3a 1a 1a

X.1    1  1  1  1  1 X.2    3 -1. A *A X.3    3 -1. *A A X.4     4. 1 -1 -1 X.5    5  1 -1. .

A = -E(5)-E(5)^4 = (1-ER(5))/2 = -b5

Irreducible representations
The irreducible representations can be computed using GAP's IrreducibleRepresentations function as follows:

gap> IrreducibleRepresentations(AlternatingGroup(5)); [ [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ], [ (1,2,3,4,5), (3,4,5) ] ->    [ [ [ 0, -1, 0 ], [ -E(5)-E(5)^4, -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4, 1 ], [ 2*E(5)+E(5)^2+E(5)^3+2*E(5)^4, 2*E(5)+E(5)^2+E(5)^3+2*E(5)^4, -1 ] ], [ [ 1, -E(5)-E(5)^4, -E(5)-E(5)^4 ], [ 0, -1, -1 ], [ 0, 1, 0 ] ] ] , [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ 1, 0, 1 ], [ -1, 0, 0 ], [ -1, -1, E(5)+E(5\ )^4 ] ], [ [ 0, -E(5)^2-E(5)^3, E(5)^2+E(5)^3 ], [ -1, -1, E(5)+E(5)^4 ], [ -E(5)-E(5)^4, -E(5)-E(5)^4, 1 ] ] ], [ (1,2,3,4,5), (3,4,5) ] -> [ [         [ E(3)^2, -1/3*E(3)-2/3*E(3)^2, E(3)^2, 4/3*E(3)+2/3*E(3)^2 ], [ E(3), 4/3*E(3)+2/3*E(3)^2, 1, -1/3*E(3)-2/3*E(3)^2 ], [ E(3), E(3), 0, -E(3)^2 ], [ E(3), -2/3*E(3)-1/3*E(3)^2, E(3), -1/3*E(3)-2/3*E(3)^2 ] ], [ [ E(3), 1/3*E(3)-1/3*E(3)^2, 0, 2/3*E(3)-2/3*E(3)^2 ], [ E(3)^2, 0, 0, 0 ], [ 0, 2/3*E(3)+1/3*E(3)^2, 1, -2/3*E(3)-1/3*E(3)^2 ], [ 1, 1, 0, -E(3) ] ] ], [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ -1, -1, -1, -1, -1 ], [ 1, 1, 0, 0, 1 ],         [ 0, -1, -1, 0, -1 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 1, 1 ] ],      [ [ -1, -1, -1, -1, -1 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1 ],          [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ] ] ] ]