Linear representation theory of symmetric group:S4

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This article discusses the linear representation theory of symmetric group:S4, a group of order 24. In the article we take $S_4$ to be the group of permutations on the set $\{ 1,2,3,4\}$.

Summary

Item Value
Degrees of irreducible representations over a splitting field 1,1,2,3,3
maximum: 3, lcm: 6, number: 5
Schur index values of irreducible representations 1,1,1,1,1
maximum: 1, lcm: 1
Smallest ring of realization for all irreducible representations (characteristic zero) $\mathbb{Z}$
Smallest field of realization for all irreducible representations, i.e., smallest splitting field (characteristic zero) $\mathbb{Q}$ (hence it is a rational representation group)
Condition for being a splitting field for this group Any field of characteristic not two or three is a splitting field.
Smallest size splitting field Field:F5, i.e., the field with five elements.

Family contexts

Family name Parameter values General discussion of linear representation theory of family
symmetric group 4 linear representation theory of symmetric groups
projective general linear group of degree two field:F3 linear representation theory of projective general linear group of degree two
COMPARE AND CONTRAST: View linear representation theory of groups of order 24 to compare and contrast the linear representation theory with other groups of order 24.

Irreducible representations

Summary information

Below is summary information on irreducible representations. Note that a particular representation may make sense, and be irreducible, only for certain kinds of fields -- see the "Values not allowed for field characteristic" and "Criterion for field" columns to see the condition the field must satisfy for the representation to be irreducible there.

Name of representation type Number of representations of this type Values not allowed for field characteristic Criterion for field What happens over a splitting field? Kernel Quotient by kernel (on which it descends to a faithful representation) Degree Schur index What happens by reducing the $\mathbb{Z}$-representation over bad characteristics?
trivial 1 -- any remains the same whole group trivial group 1 1 --
sign 1 -- any remains the same A4 in S4 (unless the characteristic is two, in which case it is the whole group) cyclic group:Z2 (unless the characteristic is two, in which case we get trivial group) 1 1 there are no bad characteristics, but it is noteworthy that in characteristic two, this becomes same as trivial representation
two-dimensional irreducible 1 3 any remains the same normal V4 in S4 symmetric group:S3 2 1 When the $\mathbb{Z}$-representation is mapped to field:F3, we get a representation that is indecomposable but not irreducible.
standard 1 2 any remains the same trivial subgroup, i.e., it is faithful symmetric group:S4 3 1 The representation is indecomposable but not irreducible. When the $\mathbb{Z}$-representation is mapped to field:F2, we get a parabolic subgroup of GL(3,2), namely $P_{2,1}$, which is isomorphic to symmetric group:S4.
product of standard and sign 1 2 any remains the same trivial subgroup, i.e., it is faithful symmetric group:S4 3 1 The representation is indecomposable but not irreducible. When the $\mathbb{Z}$-representation is mapped to field:F2, we get a parabolic subgroup of GL(3,2), namely $P_{2,1}$, which is isomorphic to symmetric group:S4. (Note the behavior is same as for the standard representation because when we go modulo 2 we forget the signs anyway)

Trivial representation

This is a one-dimensional representation sending every element of the symmetric group of degree four to the matrix $(1)$.

Sign representation

This is a one-dimensional representation that sends all even permutations to $(1)$ and all odd permutations to $(-1)$.

Degree two irreducible representation

The symmetric group of degree four has a normal subgroup of order four, namely: $\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$ (for more, see normal V4 in S4). The quotient by this subgroup is isomorphic to the symmetric group of degree three. The symmetric group of degree three has an irreducible representation of degree two that can be realized over the rationals (namely, its standard representation). Composing the quotient map with this representation of the quotient group gives an irreducible representation of degree two of the symmetric group of degree four.

Standard representation

The two irreducible representations of degree three are: the standard representation (which is the nontrivial irreducible constituent in the natural representation on a $n$-dimensional vector space via permutation of the basis elements) and the tensor product of the standard representation and the sign representation.

The natural representation given by the action on a $n$-dimensional vector space by basis permutation (here $n = 4$) has character as follows: the character of a permutation is its number of fixed points. We are decomposing this natural representation as a direct sum of the trivial representation and the standard representation. Thus, the character of the standard representation is defined as follows: the character value at a permutation is (number of fixed points) - 1.

Product of standard representation and sign representation

This is the other three-dimensional irreducible representation, and it is the tensor product of the standard representation and the sign representation. The matrix corresponding to a permutation for this representation is the sign of the permutation times the matrix corresponding to the standard representation.

Character table

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

This is the character table over characteristic zero.

Rep/Conj class $()$ (identity element) (size 1) $(1,2)$ (size 6) $(1,2,3)$ (size 8) $(1,2)(3,4)$ (size 3) $(1,2,3,4)$ (size 6)
Trivial representation 1 1 1 1 1
Sign representation 1 -1 1 1 -1
Irreducible representation of degree two with kernel of order four 2 0 -1 2 0
Standard representation 3 1 0 -1 -1
Product of standard and sign representations 3 -1 0 -1 1

Note that since all representations are realized over the rational numbers, all characters are integer-valued.

The same character table applies in any characteristic not equal to 2 or 3, where 0,-1,1,2 are interpreted, not as integers, but as elements of that field.

Here are the size-degree weighted characters (i.e., the product of the character value by the size of the conjugacy class divided by the degree of the representation).

Rep/Conj class $()$ (identity element) (size 1) $(1,2)$ (size 6) $(1,2,3)$ (size 8) $(1,2)(3,4)$ (size 3) $(1,2,3,4)$ (size 6)
Trivial representation 1 6 8 3 6
Sign representation 1 -6 8 3 -6
Irreducible representation of degree two with kernel of order four 1 0 -3 8 0
Standard representation 1 2 0 -1 -2
Product of standard and sign representations 1 -2 0 -1 2

Here is the orthogonal matrix obtained by multiplying each character value by the square root of the quotient of the size of its conjugacy class by the order of the group. Note that this is an orthogonal matrix due to the orthogonality relations between the characters.

$\begin{pmatrix} 1/\sqrt{24} & 1/2 & 1/\sqrt{3} & 1/(2\sqrt{2}) & 1/2\\ 1/\sqrt{24} & -1/2 & 1/\sqrt{3} & 1/(2\sqrt{2}) & -1/2 \\ 1/\sqrt{6} & 0 & -1/\sqrt{3} & 1/\sqrt{2} & 0 \\ 3/\sqrt{24} & 1/2 & 0 & -1/\sqrt{3} & -1/2 \\ 3/\sqrt{24} & -1/2 & 0 & -1/\sqrt{3} & 1/2 \\\end{pmatrix}$

Degrees of irreducible representations

FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:
Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroup
Size bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
Cumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization

Note that the linear representation theory of the symmetric group of degree four works over any field of characteristic not equal to two or three, and the list of degrees is $1,1,2,3,3$.

Interpretation as symmetric group

Common name of representation Degree Corresponding partition Young diagram Hook-length formula for degree Quick rule for character computation Conjugate partition Representation for conjugate partition
trivial representation 1 4 $\frac{4!}{4 \cdot 3 \cdot 2 \cdot 1}$ 1 everywhere 1 + 1 + 1 + 1 sign representation
sign representation 1 1 + 1 + 1 + 1 $\frac{4!}{4 \cdot 3 \cdot 2 \cdot 1}$ 1 on even, -1 on odd permutations 4 trivial representation
degree two irreducible representation 2 2 + 2 $\frac{4!}{3 \cdot 2 \cdot 2 \cdot 1}$ (complicated) 2 + 2 same representation because the partition is self-conjugate
standard representation 3 3 + 1 $\frac{4!}{4 \cdot 2 \cdot 1 \cdot 1}$ (number of fixed points) - 1 2 + 1 + 1 product of standard representation and sign representation
product of standard representation and sign representation 3 2 + 1 + 1 $\frac{4!}{4 \cdot 2 \cdot 1 \cdot 1}$ (sign) times (number of fixed points - 1) 3 + 1 standard representation

Interpretation as projective general linear group of degree two

Compare and contrast with linear representation theory of projective general linear group of degree two over a finite field

Below is an interpretation of the group as the projective general linear group of degree two over field:F3, the field of three elements.

Description of collection of representations Parameter for describing each representation How the representation is described Degree of each representation (general $q$ Degree of each representation ($q = 3$) Number of representations (general $q$) Number of representations ($q = 3$) Sum of squares of degrees (general $q$) Sum of squares of degrees ($q = 3$) Symmetric group name
Trivial -- $x \mapsto 1$ 1 1 1 1 1 1 trivial
Sign representation -- Kernel is projective special linear group of degree two (in this case, alternating group:A4), image is $\{ \pm 1 \}$ 1 1 1 1 1 1 sign
Nontrivial component of permutation representation of $PGL_2$ on the projective line over $\mathbb{F}_q$ -- -- $q$ 3 1 1 $q^2$ 9 standard
Tensor product of sign representation and nontrivial component of permutation representation on projective line -- -- $q$ 3 1 1 $q^2$ 9 product of sign and standard
Induced from one-dimensional representation of Borel subgroup -- -- $q + 1$ 4 $(q-3)/2$ 0 $(q+1)^2(q-3)/2$ 0 --
Unclear a nontrivial homomorphism $\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast$, with the property that $\varphi(x)^{q+1} = 1$ for all $x$, and $\varphi$ takes values other than $\pm 1$. Identify $\varphi$ and $\varphi^q$. unclear $q - 1$ 2 $(q-1)/2$ 1 $(q-1)^3/2$ 4 degree two irreducible
Total NA NA NA NA $q + 2$ 5 $q^3 - q$ 24 NA

Realizability information

Smallest ring of realization

Representation Smallest ring of realization Smallest set of elements occurring as matrix entries in the ring
trivial representation $\mathbb{Z}$ -- ring of integers $\{ 1 \}$
sign representation $\mathbb{Z}$ -- ring of integers $\{ 1, - 1 \}$
representation with kernel of order four $\mathbb{Z}$ -- ring of integers $\{ 1,0,-1 \}$
standard representation $\mathbb{Z}$ -- ring of integers $\{ 1,0,-1 \}$
product of standard and alternating representations $\mathbb{Z}$ -- ring of integers $\{ 1,0,-1 \}$

Smallest ring of realization as orthogonal matrices

Representation Smallest ring of realization
trivial representation $\mathbb{Z}$ -- ring of integers
sign representation $\mathbb{Z}$ -- ring of integers
representation with kernel of order four $\mathbb{Z}[\sqrt{3}/2]$

Character ring structure

Further information: character ring

This describes the decomposition of products of characters as sums of characters. This is:

Representation/representation trivial sign irreducible of degree two standard product of standard and sign
trivial trivial sign irreducible of degree two standard product of standard and sign
sign sign trivial irreducible of degree two product of standard and sign standard
irreducible of degree two irreducible of degree two irreducible of degree two irreducible of degree two + trivial + sign standard + product of standard and sign standard + product of standard and sign
standard standard product of standard and sign standard + product of standard and sign trivial + irreducible of degree two + standard + product of standard and sign sign + irreducible of degree two + standard + product of standard and sign
product of standard and sign product of standard and sign standard standard + product of standard and sign sign + irreducible of degree two + standard + product of standard and sign trivial + irreducible of degree two + standard + product of standard and sign

Orthogonality relations and numerical checks

General statement Verification in this case
number of irreducible representations equals number of conjugacy classes Both numbers in this case are equal to 5.
As symmetric group: Both numbers are equal to the number of unordered integer partitions of 4.
As projective general linear group $PGL(2,q), q = 3$: Both numbers are equal to $q + 2$.
sufficiently large implies splitting: if the field has characteristic not dividing the order of the group and has primitive $d^{th}$ roots of unity for $d$ the exponent of the group, it is a splitting field. In fact, for this group, any field of characteristic not 2 or 3 is a splitting field.
number of one-dimensional representations equals order of abelianization Both numbers are equal to 2: the representations are the trivial and sign representation. The derived subgroup is A4 in S4, which has index two.
sum of squares of degrees of irreducible representations equals group order $1^2 + 1^2 + 2^2 + 3^2 + 3^2 = 24 = 4!$
As symmetric group: follows from the Robinson-Schensted correspondence
As projective general linear group: See linear representation theory of projective general linear group of degree two over a finite field
degree of irreducible representation divides order of group All the degrees $1,1,2,3,3$ divide the group order which is $24$
degree of irreducible representation divides index of abelian normal subgroup No additional information conveyed since there is no nontrivial abelian normal subgroup.
row orthogonality theorem and column orthogonality theorem can be verified from the character table

Relation with quotients

Symmetric group:S4 has four normal subgroups: the whole group, the trivial subgroup, and two others. The irreducible representations with kernel a particular normal subgroup correspond precisely to the faithful irreducible representations of the quotient group; the irreducible representations with kernel containing a particular normal subgroup correspond precisely to the irreducible representations of the quotient group. Information in this regard is presented below:

Normal subgroup in whole group Normal subgroup isomorphism type Quotient group Linear representation theory of quotient group Degrees of irreducible representations of quotient group List of irreducible representations of quotient group (corresponding representation of whole group) Degrees of faithful irreducible representations of quotient group List of faithful irreducible representations of quotient group (corresponding representation of whole group)
whole group symmetric group:S4 trivial group 1 trivial (trivial) 1 trivial (trivial)
A4 in S4 alternating group:A4 cyclic group:Z2 link 1,1 trivial (trivial), sign (sign) 1 sign (sign)
normal V4 in S4 Klein four-group symmetric group:S3 link 1,1,2 trivial (trivial), sign (sign), standard (becomes two-dimensional irreducible for $S_4$, not standard for $S_4$) 2 standard (becomes two-dimensional irreducible for $S_4$, not standard for $S_4$)
trivial subgroup trivial group symmetric group:S4 link 1,1,2,3,3 trivial (trivial), sign (sign), two-dimensional irreducible (two-dimensional irreducible), standard (standard), product of standard and sign (product of standard and sign) 3,3 standard (standard), product of standard and sign (product of standard and sign)

GAP implementation

The degrees of irreducible representations can be computed using GAP's CharacterDegrees function, as follows:

gap> CharacterDegrees(SymmetricGroup(4));
[ [ 1, 2 ], [ 2, 1 ], [ 3, 2 ] ]

This means that there are two irreducible representations of degree 1, 1 of degree 2, and 2 of degree 3.

The character table of this group can be computed using GAP's CharacterTable function, as follows:

gap> Irr(CharacterTable("Symmetric",4));
[ Character( CharacterTable( "Sym(4)" ), [ 1, -1, 1, 1, -1 ] ), Character( CharacterTable( "Sym(4)" ), [ 3, -1, -1, 0, 1 ] ),
Character( CharacterTable( "Sym(4)" ), [ 2, 0, 2, -1, 0 ] ), Character( CharacterTable( "Sym(4)" ), [ 3, 1, -1, 0, -1 ] ),
Character( CharacterTable( "Sym(4)" ), [ 1, 1, 1, 1, 1 ] ) ]

The matrices of irreducible representations can be computed using GAP's IrreducibleRepresentations function, as follows:

gap> IrreducibleRepresentations(SymmetricGroup(4));
[ Pcgs([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]) ->
[ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]) ->
[ [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ],
Pcgs([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]) -> [ [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ], [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ],
[ [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, -1 ] ], [ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ], Pcgs([ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ]) ->
[ [ [ 0, -1, 0 ], [ -1, 0, 0 ], [ 0, 0, -1 ] ], [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ], [ [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, -1 ] ],
[ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ] ]

Note that this only gives the matrices of images of a generating set.