Linear representation theory of symmetric group:S4

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 information
Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2 or 3, except in the last two columns, where we consider what happens in characteristic 2 and characteristic 3.

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
This is the character table over characteristic zero.

 

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).

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

Schur functors corresponding to irreducible representations
Note that the discussion in this section relies specifically on the group being a symmetric group, and does not make sense for arbitrary finite groups.

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

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:

Degrees of irreducible representations
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.

Character table
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 ] ) ]

A visual display of the character table can be achieved as follows:

gap> Display(CharacterTable(SymmetricGroup(4))); CT1

2 3  2  3  .  2     3  1  .  .  1.

1a 2a 2b 3a 4a 2P 1a 1a 1a 3a 2b 3P 1a 2a 2b 1a 4a

X.1    1 -1  1  1 -1 X.2    3 -1 -1. 1 X.3    2. 2 -1 . X.4     3  1 -1. -1 X.5    1  1  1  1  1

Irreducible representations
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.