Standard representation of symmetric group:S4

This article discusses a three-dimensional faithful irreducible representation of symmetric group:S4, called the standard representation since it belongs to the family of standard representations of symmetric groups.

Interpretation as symmetric group
This is a faithful three-dimensional representation. One way of obtaining this representation is as follows: consider a four-dimensional vector space with basis $$e_1, e_2, e_3, e_4$$. Let the symmetric group permute the basis vectors, and consider the induced action of the symmetric group on the vector space. This is a four-dimensional representation. Consider the three-dimensional subspace of all vectors of the form $$x_1e_1 + x_2e_2 + x_3e_3 + x_4e_4$$ where $$x_1 + x_2 + x_3 + x_4 = 0$$. When the characteristic of the field is not two or three, this is a faithful, irreducible, two-dimensional representation. Note that $$e_1 - e_2$$, $$e_2 - e_3$$, and $$e_3 - e_4$$ can be taken as a basis for this, with $$e_4 - e_1$$ being the negative of the sum of these.

Interpretation as projective general linear group
The symmetric group of degree four is the projective general linear group of degree two over field:F3, and hence has an induced permutation action on the collection of one-dimensional subspaces of a two-dimensional vector space over field:F3. It turns out that there are exactly $$3 + 1 = 4$$ such subspaces, and the induced action is all permutations, so this is the same as the natural action on a set of size four. The nontrivial irreducible component of this is thus the same as the standard representation.

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:S4 into the general linear group of degree three over the field or ring. There is, however, another faithful representation of the same degree -- namely the product of the standard and sign representations, and that defines another embedding in the general linear group of degree three. The representations are inequivalent and the image subgroups are not necessarily conjugate subgroups. In particular, the embedded subgroup isomorphic to symmetric group:S4 need not be an isomorph-conjugate subgroup. (Are the subgroups obtained by these representations automorphic subgroups via the transpose-inverse map or a determinantal automorphism? Worth checking).

Further, the nature of the representation makes it clear that the representation in fact descends to an embedding of symmetric group:S4 in the projective general linear group of degree two over the field or commutative unital ring. Moreover, this latter embedding is identical both for this representation and for the product of the standard and sign representation. However, there may be other embeddings arising from projective representations.