Linear representation theory of symmetric group:S3

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This article gives specific information, namely, linear representation theory, about a particular group, namely: symmetric group:S3.
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This article discusses the representation theory of symmetric group:S3, a group of order 6. In the article we take to be the group of permutations of the set .

Summary

Item Value
Degrees of irreducible representations over a splitting field (and in particular over ) 1,1,2
maximum: 2, lcm: 2, number: 3
sum of squares: 6, quasirandom degree: 1
Schur index values of irreducible representations 1,1,1
Smallest ring of realization for all irreducible representations (characteristic zero)
Minimal splitting field, i.e., smallest field of realization for all irreducible representations (characteristic zero) (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. In particular, and are splitting fields.
Minimal splitting field in characteristic The prime field
Smallest size splitting field field:F5, i.e., the field of five elements.


Family contexts

Family name Parameter values General discussion of linear representation theory of family Section in this article Comparative note
symmetric group of degree , i.e., the group linear representation theory of symmetric groups #Interpretation as symmetric group For any symmetric group on a finite set, all irreducible linear representations can be realized with entries in , and these give irreducible representations over any field of characteristic not dividing the order of the group.
dihedral group of order and degree , i.e., the dihedral group of order six linear representation theory of dihedral groups #Interpretation as dihedral group For a dihedral group, the irreducible representations can be realized in a finite extension of but not in itself except for degrees 3,4,6 (orders 6,8,12).
general affine group of degree one over a finite field of size , i.e., field:F3, so the group is linear representation theory of general affine group of degree one over a finite field #Interpretation as general affine group of degree one
general linear group of degree two over a finite field of size , i.e., field:F2, so the group is . linear representation theory of general linear group of degree two over a finite field #Interpretation as general linear group of degree two

Related notions

Back story

This page gives information about the degrees of irreducible representations, character table, and irreducible linear representations of symmetric group:S3. It does not, however, provide an adequate explanation of how one might arrive at (i.e., deduce) the information. For more on the back story, see determination of character table of symmetric group:S3.

Irreducible representations

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.

Name of representation type Number of representations of this type Degree Schur index Criterion for field Kernel Quotient by kernel (on which it descends to a faithful representation) Characteristic 2 Characteristic 3
trivial 1 1 1 any whole group trivial group works works
sign 1 1 1 any A3 in S3 cyclic group:Z2 works, same as trivial works
standard (two-dimensional irreducible) 1 2 1 any trivial subgroup, i.e., it is faithful symmetric group:S3 works indecomposable but not irreducible


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:

Element Matrix Characteristic polynomial Minimal polynomial Trace, character value
identity element 1
1
1
1
1
1

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 , 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 in characteristic two.

Element Matrix Characteristic polynomial Minimal polynomial Trace, character value
identity element 1
1
1
-1
-1
-1

Standard representation

Further information: Standard representation of symmetric group:S3


Element Matrix for standard representation with basis , Matrix for standard representation viewed as quotient with basis Matrix for real representation as dihedral group Matrix for complex representation as dihedral group Characteristic polynomial Minimal polynomial Trace, character value Determinant
Identity element 2 1
-1 1
-1 1
0 -1
0 -1
0 -1


Since the representation is realized over , 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 .

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 three works over any field of characteristic not equal to two or three, and the list of degrees is .

Interpretation as symmetric group

Compare and contrast with linear representation theory of symmetric groups
Common name of representation Degree Corresponding partition Young diagram Hook-length formula for degree Quick rule for character computation
trivial representation 1 3 Youngdiag3.png 1 everywhere
sign representation 1 1 + 1 + 1 Youngdiag1-1-1.png 1 for even, -1 for odd
standard representation 2 2 + 1 Youngdiag2-1.png (number of fixed points of permutation) - 1

Interpretation as dihedral group

Compare and contrast with linear representation theory of dihedral groups

Below is the interpretation of the group as the dihedral group of odd degree and order .

Representation type Degree of representation Number of such representations (general ) Number of such representations (case ) List of representations
trivial representation 1 1 1 trivial representation
nontrivial representation with kernel the cyclic subgroup of order 1 1 1 sign representation
representations arising from dihedral action: a generator of the cyclic subgroup goes to a non-identity rotation of order dividing , and the elements outside get mapped to reflections 2 1 standard representation
Total -- 3 --

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

Compare and contrast with linear representation theory of general affine group of degree one over a finite field

Below is the interpretation of the group as a general affine group of degree one over the finite field with , i.e., field:F3, the field of three elements.

Representation type Degree of representation (general ) Degree of representation () Number of representations (general ) Number of representations () List of representations
one-dimensional, additive group in kernel, reduces to representation of multiplicative group 1 1 2 trivial representation, sign representation
nontrivial component of permutation representation on elements 2 1 1 standard representation
Total -- -- 3 --

Interpretation as general linear group of degree two

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

Below is the interpretation of the group as a general linear group of degree two over the finite field with , i.e., field:F2, the field of two elements.

Description of collection of representations Parameter for describing each representation How the representation is described Degree of each representation (general ) Degree of representation () Number of representations (general ) Number of representations () List of representations
One-dimensional, factor through the determinant map a homomorphism 1 1 1 trivial representation
Unclear a homomorphism , up to the equivalence , excluding the cases where unclear 1 1 sign representation
Tensor product of one-dimensional representation and the nontrivial component of permutation representation of on the projective line over a homomorphism where is the nontrivial component of permutation representation of on the projective line over 2 1 standard representation
Induced from one-dimensional representation of Borel subgroup homomorphisms with , where is treated as unordered. Induced from the following representation of the Borel subgroup: 3 0 --
Total NA NA NA NA 3

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 in characteristic zero:


Representation/Conjugacy class representative (identity element) -- size 1 (3-cycle) -- size 2 (2-transposition) -- size 3
Trivial representation 1 1 1
Sign representation 1 1 -1
Standard representation 2 -1 0

(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) (3-cycle) -- size 2 (2-transposition) -- size 3
Trivial representation 1 2 3
Sign representation 1 2 -3
Standard representation 1 -1 0

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.

Table of matrix entries

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 where 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.

Representation/element
trivial 1 1 1 1 1 1
sign 1 1 1 -1 -1 -1
standard -- top left entry 1 -1/2 -1/2 -1/2 1 -1/2
standard -- top right entry 0 0
standard -- bottom left entry 0 0
standard -- bottom right entry 1 -1/2 -1/2 1/2 -1 1/2

Realizability information

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.

Representation Smallest ring of realization Corresponding field of realization Smallest possible set of numerical values of matrix entries Comments
Trivial representation -- the ring of integers gives the trivial representation over any ring
Sign representation -- the ring of integers gives a representation over any ring; nontrivial for characteristic not equal to
Standard representation -- the ring of integers gives an irreducible representation over any ring of characteristic not equal to

Smallest ring of realization as real orthogonal matrices

Representation Smallest ring of realization with orthogonal matrices Corresponding field of realization
trivial representation
sign representation
standard representation

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.

Common name of representation Degree Square of degree Corresponding partition Young diagram Common name of corresponding Schur functor Formula for dimension of corresponding Schur functor applied to a vector space of dimension Degree of representation times this dimension Formula for calculating effect of Schur functor on character if this Schur functor is applied to any (possibly unrelated) representation whose character is
trivial representation 1 1 3 Youngdiag3.png
sign representation 1 1 1 + 1 + 1 Youngdiag1-1-1.png
standard representation 2 4 2 + 1 Youngdiag2-1.png
Total -- 6 (equals 3!, order of group) -- -- -- -- (dimension of ) --

Products and Schur functors

Character ring structure

FACTS TO CHECK AGAINST (character ring):
Sums and products: character of direct sum of linear representations is sum of characters|character of tensor product of linear representations is product of characters
Tensor products and irreducibility: tensor product of irreducible representation and one-dimensional representation is irreducible| tensor product of irreducible representations need not be irreducible

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:

Representation/representation trivial representation sign representation standard representation
trivial representation trivial sign standard
sign representation sign trivial standard
standard representation standard standard trivial + sign + standard

Schur functors

BACKGROUND INFORMATION ON Schur functors: Formula for calculating effect of Schur functor on character
Size of set being partitioned Partition for Schur functor Name of functor Formula for computing degree if original representation has degree Formula for computing character of the representation obtained after applying this functor in terms of the original character , on an element Effect on trivial representation Effect on sign representation Effect on standard representation
1 1 identity functor trivial (dim: 1) sign (dim: 1) standard (dim: 2)
2 2 symmetric square trivial (dim: 1) trivial (dim: 1) trivial + standard (dim: 3)
2 1 + 1 exterior square or alternating square empty (dim: 0) empty (dim: 0) sign (dim: 1)
3 3 symmetric cube trivial sign trivial + sign + standard (dim: 4)
3 2 + 1  ? empty (dim: 0) empty (dim: 0) standard (dim: 2)
3 1 + 1 + 1 exterior cube empty (dim: 0) empty (dim: 0) empty (dim: 0)

Group ring interpretation

Direct sum decomposition

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

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

Note that the ring of integers does not satisfy the condition for this direct sum decomposition to hold. Instead we need to use the ring (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:

where are idempotents. These are called primitive central idempotents.

Representation Degree Corresponding primitive central idempotent that gives the identity element in the corresponding direct summand How to read this from the character table
trivial representation 1 We multiply each group element by its character value, add up, and divide by the order of the group. For the trivial representation, the character values are all 1.
sign representation 1 We multiply each group element by its character value, add up, and divide by the order of the group. For the trivial representation, the character values are on the 3-cycles and the identity element and on the transpositions.
standard representation 2 We multiply each group element by its character value, add up, and divide by the order of the group. For the standard representation, the character value is 2 at the identity element, -1 at the 3-cycles, and 0 at the transpositions (so we don't need to write the transpositions).

Orthogonality relations and numerical checks

Recall that the degrees of irreducible representations are 1,1,2.

General statement Verification in this case
number of irreducible representations equals number of conjugacy classes Both numbers are equal to 3.
As symmetric group : both numbers are equal to the number of unordered integer partitions of 3.
As , : both numbers are equal to .
As : Both numbers are equal to .
sufficiently large implies splitting: if the field has characteristic not dividing the order of the group and has primitive roots of unity for 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 2: the two one-dimensional representations are the trivial and sign representations, and the abelianization is cyclic group:Z2 (arising as quotient by the subgroup of order three, which is the derived subgroup).
sum of squares of degrees of irreducible representations equals group order
degree of irreducible representation divides order of group The degrees (1,1,2) all divide the order 6.
degree of irreducible representation divides index of abelian normal subgroup The degrees (1,1,2) all divide the index 2 of the abelian normal subgroup of order 3 in the whole group.
Ito-Michler theorem The prime 3 is missing from the factors of degrees of irreducible representations, and indeed the 3-Sylow subgroup A3 in S3 is abelian and normal.
row orthogonality theorem and column orthogonality theorem Can be verified for the character table.

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.

Relation with representations of subgroups

Induced representations from subgroups

FACTS TO CHECK AGAINST (induced representation):
Degree and character: Degree of induced representation from subgroup is product of degree of original representation and index of subgroup |Character of induced representation is induced class function of character
Relation with restriction: Frobenius reciprocity
Special cases: Induced representation from regular representation of subgroup is regular representation of group|Induced representation from trivial representation of subgroup is permutation representation for action on coset space|Induced representation from trivial representation on normal subgroup factors through regular representation of quotient group
Zero values for character: Induced class function from normal subgroup is zero outside the subgroup | Induced class function from conjugacy-closed normal subgroup is index of subgroup times class function inside the subgroup and zero outside the subgroup
Subgroup Representation of subgroup Induced representation on whole group (in terms of character) Irreducible components of induced representation on whole group
3-Sylow subgroup, i.e., the alternating group trivial representation takes the value on even permutations and on odd permutations trivial representation and sign representation
3-Sylow subgroup, i.e., the alternating group nontrivial representation takes the value at the identity element, at 3-cycles, and outside standard representation
2-Sylow subgroup, such as trivial representation takes the value 3 at the identity element, 1 at each 2-transposition, and 0 at the 3-cycles trivial representation and standard representation
2-Sylow subgroup, such as sign representation takes the value 3 at the identity element, -1 at each 2-transposition, and 0 at 3-cycles sign representation and standard representation.

Restriction of representations to subgroups

Representation on whole group Subgroup Restriction (in terms of character) Irreducible components of restriction
trivial representation 3-Sylow subgroup 1 everywhere trivial representation
sign representation 3-Sylow subgroup 1 everywhere trivial representation
standard representation 3-Sylow subgroup 2 on identity, -1 on 3-cycles the two nontrivial representations
trivial representation 2-Sylow subgroup 1 everywhere trivial representation
sign representation 2-Sylow subgroup 1 on identity, -1 on non-identity element sign representation
standard representation 2-Sylow subgroup 2 on identity, 0 on non-identity element trivial representation and sign representation

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:

Representation/representation Trivial Nontrivial Nontrivial
Trivial 1 0 0
Sign 1 0 0
Standard 0 1 1

Between the whole group and its 2-Sylow subgroup:

Representation/representation Trivial Sign
Trivial 1 0
Sign 0 1
Standard 1 1

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.

GAP implementation

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 ] ] ] ]