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) | (3-regular) | (3-regular) | (3-regular) | (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 ] ) ]