Linear representation theory of symmetric group:S6

This article gives specific information, namely, linear representation theory, about a particular group, namely: symmetric group:S6.
View linear representation theory of particular groups | View other specific information about symmetric group:S6

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)	$Z$ -- ring of integers
Smallest field of realization for all irreducible representations, i.e., smallest splitting field (characteristic zero)	$Q$ -- hence it is a rational representation group
Criterion for a field to be a splitting field	Any field of characteristic not 2,3,5,7
Smallest size splitting field	field:F11

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 four works over any field of characteristic not equal to two or three, 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	</math>\frac{6!}{5 \cdot 4 \cdot 2 \cdot 1 \cdot 2 \cdot 1}</math>	2 + 2 + 1 + 1	other irreducible nine-dimensional representation
irreducible ten-dimensional 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
irreducible ten-dimensional 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

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 ] ) ]