Modular representation theory of symmetric group:S3 at 2

This article describes the modular representation theory of symmetric group:S3 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:S3 at 3.

For information on the linear representation theory in other characteristics, see linear representation theory of symmetric group:S3.

Irreducible representations
There are two irreducible representations, both of them arising from irreducible representations in characteristic zero, namely:

Trivial representation
The trivial or principal representation is a one-dimensional representation sending every element of the symmetric group to the identity matrix of order one.

Standard representation
This is obtained by computing the standard representation and then reducing modulo 2.Here is the description: consider a three-dimensional vector space with basis $$e_1, e_2, e_3$$. Let the symmetric group permute the basis vectors, and consider the induced action of the symmetric group on the vector space. This is a three-dimensional representation. Consider the two-dimensional subspace of all vectors of the form $$x_1e_1 + x_2e_2 + x_3e_3$$ where $$x_1 + x_2 + x_3 = 0$$. When the characteristic of the field is not two or three, this is a faithful, irreducible, two-dimensional representation. Note that $$e_1 - e_2$$ and $$e_2 - e_3$$ can be taken as a basis for this, with $$e_3 - e_1$$ being the negative of the sum of these.

The table below gives the matrices for the characteristic two case. Here, we convert all $$-1$$s to $$1$$s, because we are in characteristic two. Note that the map surjects to $$SL(2,2) = GL(2,2)$$.

Character table
The character table (taking values in field:F2) is as follows:

Brauer characters
Symmetric group:S3 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 standard representation.

Group ring interpretation
The group ring $$\mathbb{F}_2[S_3]$$ is not a semisimple module, and in particular, it does not break up as a direct product of matrix rings. In fact, the group ring $$K[S_3]$$ does not split up as a direct product of matrix rings even if we take $$K$$ to be the algebraic closure of $$\mathbb{F}_2$$.

GAP implementation
The character degrees can be computed using GAP's CharacterDegrees function:

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

The Brauer character table can be computed as follows:

gap> Irr(CharacterTable(SymmetricGroup(3),2)); [ Character( BrauerTable( Sym( [ 1 .. 3 ] ), 2 ), [ 1, 1 ] ), Character( BrauerTable( Sym( [ 1 .. 3 ] ), 2 ), [ 2, -1 ] ) ]