Standard representation of symmetric group:S4

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This article describes a particular irreducible linear representation for the following group: symmetric group:S4. The representation is unique up to equivalence of linear representations and is irreducible, at least over its original field of definition in characteristic zero. The representation may also be definable over other characteristics by reducing the matrices modulo that characteristic, though it may behave somewhat differently in these characteristics.
For more on the linear representation theory of the group, see linear representation theory 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.


Item Value
Degree of representation 2
Schur index 1 in all characteristics
Kernel of representation trivial subgroup, i.e., it is a faithful linear representation in all characteristics.
Quotient on which it descends to a faithful linear representation symmetric group:S4
Set of character values \{ 3,0,1,-1 \} (interpreted/reduced modulo the ring or field)
Characteristic zero: Ring generated: \mathbb{Z} -- ring of integers, Ideal within ring generated: whole ring, Field generated: \mathbb{Q} -- field of rational numbers
Rings of realization Realized over any unital ring, by composing the representation over \mathbb{Z} with the map induced by the natural homomorphism from \mathbb{Z} to the ring.
Minimal ring of realization (characteristic zero) \mathbb{Z} -- ring of integers
Minimal ring of realization in characteristic p^k The ring of integers mod p^k, \mathbb{Z}/p^k\mathbb{Z}
Minimal field of realization Prime field in all cases.
In characteristic zero, \mathbb{Q}; in characteristic p, the field \mathbb{F}_p
Size of equivalence class under automorphisms 1, because the group is a complete group, so there are no outer automorphism classes to worry about. (see symmetric groups are complete).
Size of equivalence class under Galois automorphisms 1, because the representation is realized over \mathbb{Z} in characteristic zero and more generally is realized over the prime subfield in any characteristic. See symmetric groups are rational-representation)
Size of equivalence class under action of one-dimensional representations by multiplication 2 (the other member of the equivalence class is the product of standard and sign representations
Bad characteristics 2,3 (need to verify and elaborate)

Representation table


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.


FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field):
Orthogonality relations: Character orthogonality theorem | Column orthogonality theorem
Separation results (basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions|Character determines representation in characteristic zero
Numerical facts: Characters are cyclotomic integers | Size-degree-weighted characters are algebraic integers
Character value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class| Conjugacy class of more than average size has character value zero for some irreducible character | Zero-or-scalar lemma

Character values and interpretations

Conjugacy class Size of conjugacy class (must divide the group order) Character value (must be a cyclotomic integer) Size-degree-weighted character value (must be an algebraic integer) Interpretation as symmetric group (rule: number of fixed points - 1) Philosophical justification
\{ () \} 1 3 1 The identity permutation fixes everything, so it has 4 fixed points. The character value is 4 - 1 = 3. Equals degree of representation because we are evaluating at the identity element.
\{ (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \} 3 -1 -1 The permutation has no fixed points, so the character value is 0 - 1 = -1. All the non-faithful representations have this conjugacy class in their kernel so in fact this (and its product with the sign representation) are the only representations where the character value differs from that at the identity element.
\{ (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) \} 6 1 2 The permutation fixess two points (the points not transposed, so (1,2) for instance fixes the points 3 and 4) so the character value is 2 - 1 = 1.
\{ (1,2,3,4), (1,4,3,2), (1,2,4,3), (1,3,4,2), (1,3,2,4), (1,4,2,3) \} 6 -1 -2 The permutation has no fixed points, so the character value is 0 - 1 = -1.
\{ (1,2,3), (1,3,2), (1,2,4), (1,4,2), (1,3,4), (1,4,3), (2,3,4), (2,4,3) \} 8 0 0 Each permutation has one fixed point -- the point not included in the 3-cycle (so (1,2,3) fixes 4). The character value is thus 1 - 1 = 0. This follows from irreducible character of degree greater than one takes value zero on some conjugacy class (and the fact that the character value is nonzero on all other conjugacy classes). It also follows from conjugacy class of more than average size has character value zero for some irreducible character and the fact that this conjugacy class takes nonzero values on all the non-faithful representations. Finally, it also follows from the zero-or-scalar lemma and the fact that since the conjugacy class is not central and the representation is faithful, the element cannot map to a scalar matrix.

Verification of orthonormality

General assertion Verification in this case
For an irreducible character \chi of a group G in characteristic zero, we must have \frac{1}{|G|} \sum_{g \in G} \chi(g)\overline{\chi(g)} = 1. Part of the character orthogonality theorem In this case, we have a conjugacy class of size 1 with character value 3, a conjugacy class of size 3 with character value -1, and so on. Plugging in, we get:
\frac{1}{24}[(1)(3)^2 + (3)(-1)^2 + (6)(1)^2 + (6)(-1)^2 + (8)(0)^2] = \frac{24}{24} = 1
Any two distinct irreducible characters are orthogonal. In particular, any nontrivial irreducible character \chi is orthogonal to the trivial character, so \frac{1}{|G|} \sum_{g \in G} \chi(g) = 0. Also, applying to the character of the sign representation, we get \frac{1}{|G|} \sum_{g \in G} \chi(g)\operatorname{sgn}(g) = 0 where \operatorname{sgn} denotes the sign of a permutation. In this case, we get:
\frac{1}{24}[(1)(3) + (3)(-1) + (6)(1) + (6)(-1) + (8)(0)] = 0 and \frac{1}{24}[(1)(1)(3) + (3)(1)(-1) + (6)(-1)(1) + (6)(-1)(-1) + (8)(1)(0)] = 0.
We can also verify orthogonality with the other two-dimensional representation and the other three-dimensional representation in a similar fashion.


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.