Linear representation theory of symmetric group:S4

This article discusses the linear representation theory of the symmetric group of degree four. See also linear representation theory of symmetric groups for a general discussion of the linear representation theory of all symmetric groups of finite degree.

All representations of the symmetric group of degree four can be realized over the field of rational numbers, and can in fact be realized with integer entries.

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 groups of degree two

List of irreducible representations

The trivial representation

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

The sign representation

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

The irreducible representation of degree two

The symmetric group of degree four has a normal subgroup of order four, namely: $\{(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$. 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). This gives an irreducible representation of degree two of the symmetric group of degree four.

Two irreducible representations of degree three

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 representation) and the tensor product of the standard representation and the alternating 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 alternating 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 alternating 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 $1,1,2,3,3$.

Explanation of degrees from the perspective of symmetric group of degree four

Common name of representation	Degree	Partition corresponding to representation	Hook length formula for degree	Conjugate partition	Representation for conjugate partition
trivial representation	1	4	${\frac {4!}{4\cdot 3\cdot 2\cdot 1}}$	1 + 1 + 1 + 1	sign representation
sign representation	1	1 + 1 + 1 + 1	${\frac {4!}{4\cdot 3\cdot 2\cdot 1}}$	4	trivial representation
degree two irreducible representation	2	2 + 2	${\frac {4!}{3\cdot 2\cdot 2\cdot 1}}$	2 + 2	same representation because the partition is self-conjugate
standard representation	3	3 + 1 (or is it 1 + 3?)	${\frac {4!}{4\cdot 2\cdot 2\cdot 1}}$	1 + 3 (or is it 3 + 1?)	product of standard representation and sign representation
product of standard representation and sign representation	3	1 + 3 (or is it 3 + 1?)	${\frac {4!}{4\cdot 2\cdot 2\cdot 1}}$	3 + 1 (or is it 1 + 3?)	standard representation

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

GAP implementation

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.