Modular representation theory of symmetric group:S4 at 2

This article describes the modular representation theory of symmetric group:S4 at the prime two, i.e., for fields of characteristic two, specifically field:F2 and its extensions.

For information on the linear representation theory in characteristic three (the other modular case) see modular representation theory of symmetric group:S4 at 3.

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.

Trivial representation
This is a representation that sends every element to the matrix $$( 1 )$$.

Two-dimensional irreducible representation
This representation is described as follows: the group symmetric group:S4 has a normal subgroup $$\{, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$$ (see normal V4 in S4). The quotient group is isomorphic to symmetric group:S3. The quotient has an irreducible two-dimensional representation over field:F2, namely its standard representation. Composing with the quotient map, we get a two-dimensional irreducible representation of the original group symmetric group:S4.

Character table
The character table is as follows:

Brauer characters
Symmetric group:S4 is a rational representation group, in the sense that all its representations in characteristic zero can be realized using matrices with integer entries, and hence over the rational numbers. On the other side, all irreducible representations in characteristic two are realized over field:F2.

However, the eigenvalues of these representations do not live over field:F2, i.e., the matrices are not diagonalizable over this field. In order to diagonalize, we need to adjoin cube roots of unity on both sides. In other words, we need to fix a bijection between cube roots of unity in a quadratic extension of field:F2 and cube roots of unity in a quadratic extension of the rational numbers. Since the actual matrix entries and character values all live in the base fields, the Brauer characters ultimately will turn out not to depend on this choice of bijection.

Brauer character table
There are two 2-regular conjugacy classes: the identity element and the 3-cycles. There are two Brauer characters: the character of the trivial representation and the character of the two-dimensional irreducible representation.

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

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

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

gap> Irr(CharacterTable(SymmetricGroup(4),2)); [ Character( BrauerTable( Sym( [ 1 .. 4 ] ), 2 ), [ 1, 1 ] ), Character( BrauerTable( Sym( [ 1 .. 4 ] ), 2 ), [ 2, -1 ] ) ]