Linear representation theory of symmetric group:S5

This article gives specific information, namely, linear representation theory, about a particular group, namely: symmetric group:S5.
View linear representation theory of particular groups | View other specific information about symmetric group:S5

This article describes the linear representation theory of symmetric group:S5, a group of order $120$. We take this to be the group of permutations on the set $\{1,2,3,4,5 \}$.

Summary

Item	Value
Degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$)	1,1,4,4,5,5,6 maximum: 6, lcm: 60, number: 7, sum of squares: 120
Schur index values of irreducible representations	1,1,1,1,1,1,1 maximum: 1, lcm: 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 equal to 2,3, or 5.
Smallest size splitting field	field:F7, i.e., the field of 7 elements.

Family contexts

Family name	Parameter values	General discussion of linear representation theory of family
symmetric group	5	linear representation theory of symmetric groups
projective general linear group of degree two	field:F5	linear representation theory of projective general linear group of degree two

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 $1,1,4,4,5,5,6$.

Interpretation as symmetric group

Common name of representation	Degree	Corresponding partition	Young diagram	Hook-length formula for degree	Conjugate partition	Representation for conjugate partition
trivial representation	1	5		$\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$	1 + 1 + 1 + 1 + 1	sign representation
sign representation	1	1 + 1 + 1 + 1 + 1		$\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$	5	trivial representation
standard representation	4	4 + 1		$\frac{5!}{5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$	2 + 1 + 1 + 1	product of standard and sign representation
product of standard and sign representation	4	2 + 1 + 1 + 1		$\frac{5!}{5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$	4 + 1	standard representation
irreducible five-dimensional representation	5	3 + 2		$\frac{5!}{4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$	2 + 2 + 1	other irreducible five-dimensional representation
irreducible five-dimensional representation	5	2 + 2 + 1		$\frac{5!}{4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$	3 + 2	other irreducible five-dimensional representation
exterior square of standard representation	6	3 + 1 + 1		$\frac{5!}{5 \cdot 2 \cdot 2 \cdot 1 \cdot 1}$	3 + 1 + 1	the same representation, because the partition is self-conjugate.

Interpretation as projective general linear group of degree two

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

Below is an interpretation of the group as the projective general linear group of degree two over field:F5, the field of five elements.

Description of collection of representations	Parameter for describing each representation	How the representation is described	Degree of each representation (general odd $q$)	Degree of each representation ($q = 5$)	Number of representations (general odd $q$)	Number of representations ($q = 5$)	Sum of squares of degrees (general odd $q$)	Sum of squares of degrees ($q = 5$)	Symmetric group name
Trivial	--	$x \mapsto 1$	1	1	1	1	1	1	trivial
Sign representation	--	Kernel is projective special linear group of degree two (in this case, alternating group:A5), image is $\{ \pm 1 \}$	1	1	1	1	1	1	sign
Nontrivial component of permutation representation of $PGL_2$ on the projective line over $\mathbb{F}_q$	--	--	$q$	5	1	1	$q^2$	25	irreducible 5D
Tensor product of sign representation and nontrivial component of permutation representation on projective line	--	--	$q$	5	1	1	$q^2$	25	other irreducible 5D
Induced from one-dimensional representation of Borel subgroup	?	?	$q + 1$	6	$(q-3)/2$	1	$(q+1)^2(q-3)/2$	36	exterior square of standard representation
Unclear	a nontrivial homomorphism $\varphi:\mathbb{F}_{q^2}^\ast \to \mathbb{C}^\ast$, with the property that $\varphi(x)^{q+1} = 1$ for all $x$, and $\varphi$ takes values other than $\pm 1$. Identify $\varphi$ and $\varphi^q$.	unclear	$q - 1$	4	$(q-1)/2$	2	$(q-1)^3/2$	32	standard representation, product of standard and sign
Total	NA	NA	NA	NA	$q + 2$	7	$q^3 - q$	120	NA

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

Representation/conjugacy class representative and size	$()$ (size 1)	$(1,2)$ (size 10)	$(1,2)(3,4)$ (size 15)	$(1,2,3)$ (size 20)	$(1,2,3)(4,5)$ (size 20)	$(1,2,3,4,5)$ (size 24)	$(1,2,3,4)$ (size 30)
trivial representation	1	1	1	1	1	1	1
sign representation	1	-1	1	1	-1	1	-1
standard representation	4	2	0	1	-1	-1	0
product of standard and sign representation	4	-2	0	1	1	-1	0
irreducible five-dimensional representation	5	1	1	-1	1	0	-1
irreducible five-dimensional representation	5	-1	1	-1	-1	0	1
exterior square of standard representation	6	0	-2	0	0	1	0

Below are the size-degree-weighted characters, i.e., these are obtained by multiplying the character value by the size of the conjugacy class and then dividing by the degree of the representation. Note that size-degree-weighted characters are algebraic integers.

Representation/conjugacy class representative and size	$()$ (size 1)	$(1,2)$ (size 10)	$(1,2)(3,4)$ (size 15)	$(1,2,3)$ (size 20)	$(1,2,3)(4,5)$ (size 20)	$(1,2,3,4,5)$ (size 24)	$(1,2,3,4)$ (size 30)
trivial representation	1	10	15	20	20	24	30
sign representation	1	-10	15	20	-20	24	-30
standard representation	1	5	0	5	-5	-6	0
product of standard and sign representation	1	-5	0	5	5	-6	0
irreducible five-dimensional representation	1	2	3	-4	4	0	-6
irreducible five-dimensional representation	1	-2	3	-4	-4	0	6
exterior square of standard representation	1	0	-5	0	0	4	0

GAP implementation

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

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

This means that there are 2 degree 1 irreducible representations, 2 degree 4 irreducible representations, 2 degree 5 irreducible representations, and 1 degree 6 irreducible representation.

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

gap> Irr(CharacterTable(SymmetricGroup(5)));
[ Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 1, -1, 1, 1, -1, -1, 1 ] ),
  Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 4, -2, 0, 1, 1, 0, -1 ] ),
  Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 5, -1, 1, -1, -1, 1, 0 ] ),
  Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 6, 0, -2, 0, 0, 0, 1 ] ),
  Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 5, 1, 1, -1, 1, -1, 0 ] ),
  Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 4, 2, 0, 1, -1, 0, -1 ] ),
  Character( CharacterTable( Sym( [ 1 .. 5 ] ) ), [ 1, 1, 1, 1, 1, 1, 1 ] ) ]