Linear representation theory of symmetric group:S3

This article discusses the representation theory of symmetric group:S3, a group of order 6. In the article we take $$S_3$$ to be the group of permutations of the set $$\{ 1,2,3 \}$$.

Related notions

 * Modular representation theory of symmetric group:S3 at 2: The representation theory over field:F2 and in other fields of characteristic two.
 * Modular representation theory of symmetric group:S3 at 3: The representation theory over field:F3 and in other fields of characteristic three.
 * Projective representation theory of symmetric group:S3

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
The trivial or principal representation is a one-dimensional representation sending every element of the symmetric group to the identity matrix of order one. This representation makes sense over all fields, and its character is 1 on all elements:

Sign representation
The sign representation is a one-dimensional representation sending every permutation to its sign: the even permutations get sent to 1 and the odd permutations get sent to -1. The kernel of this representation (i.e. the permutations that get sent to one) is the alternating group: the unique cyclic subgroup of order three comprising permutations $$(1,2,3)$$, $$(1,3,2)$$ and the identity permutation. The three permutations of order two all get sent to -1.

This representation makes sense over any field, but when the characteristic of the field is two, it is the same as the trivial representation, because $$1 = -1$$ in characteristic two.

Standard representation
Since the representation is realized over $$\mathbb{Z}$$, it makes sense over all characteristics. The only characteristic where it is not irreducible is characteristic 3. In characteristic 3, the representation is indecomposable but not irreducible.

Here is an alternative perspective on this representation in characteristic 3. The symmetric group is identified with the general affine group of degree one over the field of three elements. In other words, it is the semidirect product of the additive group of this field (a cyclic group of order three) and the multiplicative group of this field, where the multiplicative group acts on the additive group by multiplication. Via the general fact that embeds a general affine group in a general linear group of one size higher, we get a faithful representation of the symmetric group on three elements in the general linear group of degree two over field:F3, i.e., in $$GL(2,3)$$.

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

Interpretation as dihedral group
Below is the interpretation of the group as the dihedral group of odd degree $$n = 3$$ and order $$2n = 6$$.

For more information on how the standard representation corresponds to the dihedral action, see the discussion of standard representation earlier on this page.

Interpretation as general affine group of degree one
Below is the interpretation of the group as a general affine group of degree one over the finite field $$\mathbb{F}_q$$ with $$q = 3$$, i.e., field:F3, the field of three elements.

Interpretation as general linear group of degree two
Below is the interpretation of the group as a general linear group of degree two over the finite field $$\mathbb{F}_q$$ with $$q = 2$$, i.e., field:F2, the field of two elements.

Character table
This is the character table in 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).

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.

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

Using real orthogonal matrices as dihedral group
This table satisfies the grand orthogonality theorem -- in particular, any two rows are orthogonal and each row has norm $$1/n$$ where $$n$$ is the degree of the representation. Note that unlike the character table, this table is not canonical and depends on the specific choice of matrices used for the two-dimensional representation.

Smallest ring of realization
Here are the representations and the smallest rings over which they can be realized. A representation that can be realized over a ring can be realized over any field containing a homomorphic image of that ring. In particular, a representation that can be realized over the ring of integers can be realized over any ring.

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. Note that the product of characters of two representations is realized as the character of the tensor product of these representations. This is as follows:

Direct sum decomposition
If $$K$$ is any field whose characteristic is not 2 or 3, then the group ring $$K[S_3]$$ splits as a direct sum of two-sided ideals corresponding to the irreducible representations:

$$K[S_3] \cong M_1(K) \oplus M_1(K) \oplus M_2(K) = K \oplus K \oplus M_2(K)$$

More generally, if $$R$$ is any commutative unital ring that is uniquely 2-divisible and uniquely 3-divisible, then we can write:

$$R[S_3] \cong M_1(R) \oplus M_1(R) \oplus M_2(R) = R \oplus R \oplus M_2(R)$$

Note that the ring of integers $$\mathbb{Z}$$ does not satisfy the condition for this direct sum decomposition to hold. Instead we need to use the ring $$\mathbb{Z}[1/2,1/3]$$ (In general, we need to use a ring that is uniquely divisible by all primes dividing the order of the group).

Explicit decomposition and idempotents
We can write:

$$R[S_3] = M_1(R)e_1 \oplus M_1(R)e_2 \oplus M_2(R)e_3$$

where $$e_1,e_2,e_3$$ are idempotents. These are called primitive central idempotents.

Orthogonality relations and numerical checks
Recall that the degrees of irreducible representations are 1,1,2.

Action of automorphisms
The automorphism group preserves each irreducible representation. This can be explained by the fact that every automorphism is inner, since the group is complete.

Relationship between irreducibles and those of subgroups: Frobenius reciprocity
Here, the rows correspond to irreducible representations of the whole group, and the columns correspond to irreducible representations of the subgroup. The number in a cell is the multiplicity of the column representation in the restriction of the row representation to the subgroup; equivalently, it is the multiplicity of the row representation in the induced representation from the column representation of the subgroup to the whole group. These numbers are equal by Frobenius reciprocity.

Between the whole group and its 3-Sylow subgroup:

Between the whole group and its 2-Sylow subgroup:

Verification of the McKay conjecture
The McKay conjecture needs to be verified for primes 2 and 3. Since the 3-Sylow subgroup is normal, nothing needs to be checked for 3. The 2-Sylow subgroup is self-normalizing. The two numbers are:


 * 1) The number of odd-dimensional characters of the symmetric group: This is 2.
 * 2) The number of odd-dimensional characters of the 2-Sylow subgroup: This is 2.

Hence, the McKay conjecture is true for this group.

Degrees of irreducible representations
The degrees of irreducible representations can be computed using the CharacterDegrees function:

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

Character table
The character table can be computed using the Irr and CharacterTable functions:

gap> Irr(CharacterTable(SymmetricGroup(3))); [ Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 1, -1, 1 ] ), Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 2, 0, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 3 ] ) ), [ 1, 1, 1 ] ) ]

A nicer display can be achieved using the Display function:

gap> Display(CharacterTable(SymmetricGroup(3))); CT1

2 1  1  .     3  1  .  1

1a 2a 3a 2P 1a 1a 3a 3P 1a 2a 1a

X.1    1 -1  1 X.2    2. -1 X.3    1  1  1

Irreducible representations
The irreducible representations can be computed using the IrreducibleRepresentations function:

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