# Linear representation theory of symmetric group:S6

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This article describes the linear representation theory of symmetric group:S6, a group of order $720$. We take this to be the group of permutations on the set $\{ 1,2,3,4,5,6 \}$.

## Summary

Item Summary
Degrees of irreducible representations over a splitting field 1,1,5,5,5,5,9,9,10,10,16
maximum: 16, lcm: 720, number: 11, sum of squares: 720
Schur index values of irreducible representations 1,1,1,1,1,1,1,1,1,1,1
Smallest ring of realization for all irreducible representations (characteristic zero) $\mathbb{Z}$ -- ring of integers
Smallest field of realization for all irreducible representations, i.e., smallest splitting field (characteristic zero) $\mathbb{Q}$ -- hence it is a rational representation group
Criterion for a field to be a splitting field Any field of characteristic not 2,3, or 5
Smallest size splitting field field:F7

## Family contexts

Family name Parameter value General discussion of linear representation theory of family
symmetric group 6 linear representation theory of symmetric groups

## 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 six works over any field of characteristic not equal to 2, 3, or 5, and the list of degrees is $1,1,5,5,5,5,9,9,10,10,16$.

### Interpretation as symmetric group

Common name of representation Degree Partition corresponding to representation Hook length formula for degree Conjugate partition Representation for conjugate partition
trivial representation 1 6 $\frac{6!}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$ 1 + 1 + 1 + 1 + 1 + 1 sign representation
sign representation 1 1 + 1 + 1 + 1 + 1 + 1 $\frac{6!}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$ 6 trivial representation
standard representation 5 5 + 1 $\frac{6!}{6 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ 2 + 1 + 1 + 1 + 1 product of standard and sign representation
product of standard and sign representation 5 2 + 1 + 1 + 1 + 1 $\frac{6!}{6 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ 5 + 1 standard representation
irreducible five-dimensional representation 5 3 + 3 $\frac{6!}{4 \cdot 3 \cdot 3 \cdot 2 \cdot 2 \cdot 1}$ 2 + 2 + 2 other irreducible five-dimensional representation
irreducible five-dimensional representation 5 2 + 2 + 2 $\frac{6!}{4 \cdot 3 \cdot 2 \cdot 3 \cdot 2 \cdot 1}$ 3 + 3 other irreducible five-dimensional representation
irreducible nine-dimensional representation 9 4 + 2 $\frac{6!}{5 \cdot 4 \cdot 2 \cdot 1 \cdot 2 \cdot 1}$ 2 + 2 + 1 + 1 other irreducible nine-dimensional representation
irreducible nine-dimensional representation 9 2 + 2 + 1 + 1 $\frac{6!}{5 \cdot 4 \cdot 2 \cdot 1 \cdot 2 \cdot 1}$ 2 + 2 + 1 + 1 other irreducible nine-dimensional representation
second exterior power of the standard representation 10 4 + 1 + 1 $\frac{6!}{6 \cdot 3 \cdot 2 \cdot 1 \cdot 2 \cdot 1}$ 3 + 1 + 1 + 1 other irreducible ten-dimensional representation
third exterior power of the standard representation 10 3 + 1 + 1 + 1 $\frac{6!}{6 \cdot 2 \cdot 1 \cdot 3 \cdot 2 \cdot 1}$ 4 + 1 + 1 other irreducible ten-dimensional representation
irreducible sixteen-dimensional representation 16 3 + 2 + 1 $\frac{6!}{5 \cdot 3 \cdot 1 \cdot 3 \cdot 1 \cdot 1}$ 3 + 2 + 1 same, i.e., self-conjugate

## 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

The character table below is incomplete, it contains only eight of the eleven representations.

Representation/conjugacy class representative and size $()$ (size 1) $(1,2)$ (size 15) $(1,2)(3,4)(5,6)$ (size 15) $(1,2,3)$ (size 40) $(1,2,3)(4,5,6)$ (size 40) $(1,2)(3,4)$ (size 45) $(1,2,3,4)$ (size 90) $(1,2,3,4)(5,6)$ (size 90) $(1,2,3)(4,5)$ (size 120) $(1,2,3,4,5,6)$ (size 120) $(1,2,3,4,5)$ -- size 144
trivial 1 1 1 1 1 1 1 1 1 1 1
sign 1 -1 -1 1 1 1 -1 1 -1 -1 1
standard 5 3 -1 2 -1 1 1 -1 0 -1 0
product of standard and sign 5 -3 1 2 -1 1 -1 -1 0 1 0
other five-dimensional irreducible 5 -1 3 -1 2 1 1 -1 -1 0 0
other five-dimensional irreducible 5 1 -3 -1 2 1 -1 -1 1 0 0
second exterior power of standard 10 2 -2 1 1 -2 0 0 -1 1 0
third exterior power of standard 10 -2 2 1 1 -2 0 0 1 -1 0

## GAP implementation

The degrees of irreducible representations can be computed using GAP's CharacterDegrees function:

gap> CharacterDegrees(SymmetricGroup(6));
[ [ 1, 2 ], [ 5, 4 ], [ 9, 2 ], [ 10, 2 ], [ 16, 1 ] ]

This means there are 2 degree 1 irreducible representations, 4 degree 5 irreducible representations, 2 degree 9 irreducible representations, 2 degree 10 irreducible representations, and 1 degree 16 irreducible representation.

The characters of irreducible representations can be computed in full using GAP's CharacterTable function:

gap> Irr(CharacterTable(SymmetricGroup(6)));
[ Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 1, -1, 1, -1, 1, -1, 1, -1, 1,
1, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ),
[ 5, -3, 1, 1, 2, 0, -1, -1, -1, 0, 1 ] ), Character( CharacterTable( Sym(
[ 1 .. 6 ] ) ), [ 9, -3, 1, -3, 0, 0, 0, 1, 1, -1, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 5, -1, 1, 3, -1, -1, 2, 1, -1,
0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ),
[ 10, -2, -2, 2, 1, 1, 1, 0, 0, 0, -1 ] ), Character( CharacterTable( Sym(
[ 1 .. 6 ] ) ), [ 16, 0, 0, 0, -2, 0, -2, 0, 0, 1, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 5, 1, 1, -3, -1, 1, 2, -1, -1,
0, 0 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ),
[ 10, 2, -2, -2, 1, -1, 1, 0, 0, 0, 1 ] ), Character( CharacterTable( Sym(
[ 1 .. 6 ] ) ), [ 9, 3, 1, 3, 0, 0, 0, -1, 1, -1, 0 ] ),
Character( CharacterTable( Sym( [ 1 .. 6 ] ) ), [ 5, 3, 1, -1, 2, 0, -1, 1, -1,
0, -1 ] ), Character( CharacterTable( Sym( [ 1 .. 6 ] ) ),
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ) ]