Modular representation theory of symmetric group:S4 at 3

This article describes the modular representation theory of symmetric group:S4 at the prime three, i.e., for fields of characteristic three, specifically field:F3 and its extensions.

For information on the linear representation theory in characteristic two (the other modular case) see modular representation theory of symmetric group:S4 at 2.

For information on the linear representation theory in other characteristics (including characteristic zero, the typical case), see linear representation theory of symmetric group:S4.

Summary information
Below is summary information on irreducible representations.

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

gap> CharacterDegrees(SymmetricGroup(4),3); [ [ 1, 2 ], [ 3, 2 ] ]

The Brauer character table can be computed using GAP's CharacterTable function:

gap> Irr(CharacterTable(SymmetricGroup(4),3)); [ Character( BrauerTable( Sym( [ 1 .. 4 ] ), 3 ), [ 1, -1, 1, -1 ] ), Character( BrauerTable( Sym( [ 1 .. 4 ] ), 3 ), [ 1, 1, 1, 1 ] ), Character( BrauerTable( Sym( [ 1 .. 4 ] ), 3 ), [ 3, -1, -1, 1 ] ), Character( BrauerTable( Sym( [ 1 .. 4 ] ), 3 ), [ 3, 1, -1, -1 ] ) ]