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 .
|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.|
- 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
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|
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|
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 math>1 = -1</math> in characteristic two.
|Element||Matrix||Characteristic polynomial||Minimal polynomial||Trace, character value|
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|
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||1 everywhere|
|sign representation||1||1 + 1 + 1||1 for even, -1 for odd|
|standard representation||2||2 + 1||(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|
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
|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|
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
|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||--|
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|
(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|
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.
|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|
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|
Products and Schur functors
Character ring structure
Further information: character ring
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|
|standard representation||standard||standard||trivial + sign + standard|
|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
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
|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 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 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:
- The number of odd-dimensional characters of the symmetric group: This is 2.
- The number of odd-dimensional characters of the 2-Sylow subgroup: This is 2.
Hence, the McKay conjecture is true for this group.