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.

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

Rep/Conj class	$()$ (identity element)	$(1,2)$	$(1,2,3)$	$(1,2)(3,4)$	$(1,2,3,4)$
Trivial representation	1	1	1	1	1
Sign representation	1	-1	1	1	-1
Representation 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

Degrees of irreducible representations

Over characteristic not equal to two or three, the degrees of irreducible representations are $1,1,2,3,3$.

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	$\mathbb {Z} [{\sqrt {3}}/2]$

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.