Modular representation theory of symmetric group:S4 at 3

This article gives specific information, namely, modular representation theory, about a particular group, namely: symmetric group:S4.
View modular representation theory of particular groups | View other specific information about symmetric group:S4

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.

Irreducible representations

Summary information

Below is summary information on irreducible representations.

Name of representation type Number of representations of this type Field of realization Kernel Degree Liftable to ordinary representation (characteristic zero)?
trivial 1 field:F3 whole group 1 Yes
sign 1 field:F3 A4 in S4 1 Yes
standard 1 field:F3 trivial subgroup 3 Yes
product of standard and sign 1 field:F3 trivial subgroup 3 Yes

Character table

Rep/conj class $()$ (3-regular) $(1,2,3,4)$ (3-regular) $(1,2)(3,4)$ (3-regular) $(1,2)$ (3-regular) $(1,2,3)$ (not 3-regular)
trivial 1 1 1 1 1
sign 1 -1 1 -1 1
standard 0 -1 -1 1 0
product of standard and sign 0 1 -1 -1 0

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 ] ) ]