Standard representation of symmetric group:S3

This article discusses a two-dimensional faithful irreducible representation of symmetric group:S3, called the standard representation since it belongs to the family of standard representations of symmetric groups.

Representation table
The representation can be defined in many equivalent ways. Note that the first two description columns show that the representation can be realized over $$\mathbb{Z}$$. In particular, this allows us to define it over any field and more generally over any unital ring by composing with the homomorphism from $$\mathbb{Z}$$ to that ring. In characteristic 3, however, the representation is not irreducible (as discussed later).

 

Interpretation as symmetric group
This is a faithful two-dimensional representation. One way of obtaining this representation is as follows: 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.

Here are the details on how the matrices are computed.

Interpretation as dihedral group
Over the real numbers, this representation is conjugate to the representation as orthogonal matrices, where we view the symmetric group of degree three as the dihedral group acting on three elements. Here, the $$3$$-cycles act as rotations by multiples of $$2\pi/3$$, and the transpositions act as reflections about suitable axes.

Character values and interpretations
The character can be computed using any of the interpretations provided. See below:

Embeddings in general linear groups and projective general linear groups
For any field (and more generally, any commutative unital ring), this faithful representation defines an embedding of symmetric group:S3 into the general linear group of degree two over the field or ring. Further, because this is the only faithful representation of degree two (up to equivalence) the image of symmetric group:S3 is an isomorph-conjugate subgroup in the target group. When the characteristic of the field (or more generally, the prime-base logarithm of the characteristic of the ring) is not 3, we get an irreducible representation. In characteristic 3, we get an indecomposable linear representation that is not irreducible.

Further, the nature of the representation makes it clear that the representation in fact descends to an embedding of symmetric group:S3 in the projective general linear group of degree two over the field or commutative unital ring. Note, however, that it is not necessary that this embedding be as an isomorph-conjugate subgroup, because $$S_3$$ may have other projective representations. For more, see projective representation theory of symmetric group:S3.

We consider the special case of finite fields and elaborate on the embeddings below:

Next, we consider some examples of finite rings that are not fields:

Construction of representation as homomorphism
gap> G := SymmetricGroup(3);; phi := Filtered(IrreducibleRepresentations(G),x -> Length(Identity(G)^x) = 2)[1]; Pcgs([ (2,3), (1,2,3) ]) -> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ]

The images of all the elements under the representation can be printed explicitly as follows:

gap> List(G, x -> [x,x^phi]); [ [, [ [ 1, 0 ], [ 0, 1 ] ] ], [ (1,3), [ [ 0, E(3) ], [ E(3)^2, 0 ] ] ], [ (1,2,3), [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ], [ (2,3), [ [ 0, 1 ], [ 1, 0 ] ] ], [ (1,3,2), [ [ E(3)^2, 0 ], [ 0, E(3) ] ] ], [ (1,2), [ [ 0, E(3)^2 ], [ E(3), 0 ] ] ] ]

Construction of character
gap> Filtered(Irr(CharacterTable(SymmetricGroup(3))),x -> DegreeOfCharacter(x) = 2)[1]; Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 2, 0, -1 ] )