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.

Summary

Item Value
Degree of representation 3
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 (interpreted/reduced modulo the ring or field)
Characteristic zero: Ring generated: -- ring of integers, Ideal within ring generated: whole ring, Field generated: -- field of rational numbers
Rings of realization Realized over any unital ring, by composing the representation over with the map induced by the natural homomorphism from to the ring.
Minimal ring of realization (characteristic zero) -- ring of integers
Minimal set of values in matrix entries realizing this representation
Minimal ring of realization in characteristic The ring of integers mod ,
Minimal field of realization Prime field in all cases.
In characteristic zero, ; in characteristic , the field
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 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

Cycle decomposition notation One-line notation, i.e., image of string Matrix for action with basis Characteristic polynomial Minimal polynomial Trace, character value Determinant
1234 3 1
1243 = 0 -1
1324 = 0 -1
1342 0 1
1423 0 1
1432 = 0 -1
2134 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] = 0 -1
2143 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] = -1 1
2314 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] 0 1
2341 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] -1 -1
2413 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] -1 -1
2431 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] 0 1
3124 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] 0 1
3142 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] -1 -1
3214 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] = 0 -1
3241 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] 0 1
3412 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] = -1 1
3421 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] -1 -1
4123 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] -1 -1
4132 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] 0 1
4213 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] 0 1
4231 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] = 0 -1
4312 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] -1 -1
4321 PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] = -1 1


Interpretation as symmetric group

For more on the general context, see standard representation

This is a faithful three-dimensional representation. One way of obtaining this representation is as follows: consider a four-dimensional vector space with basis . 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 where . When the characteristic of the field is not two or three, this is a faithful, irreducible, two-dimensional representation. Note that , , and can be taken as a basis for this, with being the negative of the sum of these.

Interpretation as projective general linear group

For more on the general context, see nontrivial irreducible component of permutation representation of projective general linear group of degree two on projective line

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 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.

Character

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) Interpretation as projective general linear group (rule: number of fixed one-dimensional subspaces - 1) over field:F3 (so subspaces in total) Philosophical justification
1 3 1 The identity permutation fixes everything, so it has 4 fixed points. The character value is . Fixes all 4 subspaces, so character value is . Equals degree of representation because we are evaluating at the identity element.
3 -1 -1 The permutation has no fixed points, so the character value is . The element of has no eigenvalues over field:F3 (it is in the torus corresponding to the extension ), so it has no fixed subspaces. The character value is thus . 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.
6 1 2 The permutation fixess two points (the points not transposed, so for instance fixes the points and ) so the character value is . The element in is diagonalizable with two distinct eigenvalues, so it has precisely two fixed subspaces -- the eigenspaces. The character value is .
6 -1 -2 The permutation has no fixed points, so the character value is . The element of has no eigenvalues over field:F3 (it is in the torus corresponding to the extension ), so it has no fixed subspaces. The character value is thus .
8 0 0 Each permutation has one fixed point -- the point not included in the 3-cycle (so fixes ). The character value is thus . The element of arises from a unipotent of the form . This has one one-dimensional eigenspace, so the character value is . 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 of a group in characteristic zero, we must have . 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:
Any two distinct irreducible characters are orthogonal. In particular, any nontrivial irreducible character is orthogonal to the trivial character, so . Also, applying to the character of the sign representation, we get where denotes the sign of a permutation. In this case, we get:
and .
We can also verify orthogonality with the other two-dimensional representation and the other three-dimensional representation in a similar fashion.

Embedding

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.