Linear representation theory of symmetric group:S6

This article describes the linear representation theory of symmetric group:S6, a group of order $$720$$. We take this to be the group of permutations on the set $$\{ 1,2,3,4,5,6 \}$$.

Degrees of irreducible representations
Note that the linear representation theory of the symmetric group of degree six works over any field of characteristic not equal to 2, 3, or 5, and the list of degrees is $$1,1,5,5,5,5,9,9,10,10,16$$.

Character table
 The character table below is incomplete, it contains only eight of the eleven representations.



GAP implementation
The degrees of irreducible representations can be computed using GAP's CharacterDegrees function:

gap> CharacterDegrees(SymmetricGroup(6)); [ [ 1, 2 ], [ 5, 4 ], [ 9, 2 ], [ 10, 2 ], [ 16, 1 ] ]

This means there are 2 degree 1 irreducible representations, 4 degree 5 irreducible representations, 2 degree 9 irreducible representations, 2 degree 10 irreducible representations, and 1 degree 16 irreducible representation.

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

gap> Irr(CharacterTable(SymmetricGroup(6))); [ Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 1, -1, 1, -1, 1, -1, 1, -1, 1,     1, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ),    [ 5, -3, 1, 1, 2, 0, -1, -1, -1, 0, 1 ] ), Character( CharacterTable( Sym(    [ 1 .. 6 ] ) ), [ 9, -3, 1, -3, 0, 0, 0, 1, 1, -1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 5, -1, 1, 3, -1, -1, 2, 1, -1,     0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ),    [ 10, -2, -2, 2, 1, 1, 1, 0, 0, 0, -1 ] ), Character( CharacterTable( Sym(    [ 1 .. 6 ] ) ), [ 16, 0, 0, 0, -2, 0, -2, 0, 0, 1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 5, 1, 1, -3, -1, 1, 2, -1, -1,     0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ),    [ 10, 2, -2, -2, 1, -1, 1, 0, 0, 0, 1 ] ), Character( CharacterTable( Sym(    [ 1 .. 6 ] ) ), [ 9, 3, 1, 3, 0, 0, 0, -1, 1, -1, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 5, 3, 1, -1, 2, 0, -1, 1, -1,     0, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ),    [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ) ]