# Difference between revisions of "Linear representation theory of alternating group:A5"

View linear representation theory of particular groups | View other specific information about alternating group:A5

## Summary

Item Value
degrees of irreducible representations over a splitting field 1,3,3,4,5
maximum: 5, lcm: 60, number: 5, sum of squares: 60

## Family contexts

Family name Parameter values General discussion of linear representation theory of family
alternating group 5 linear representation theory of alternating groups
projective general linear group of degree two over a finite field field:F4 linear representation theory of projective general linear group of degree two over a finite field
projective special linear group of degree two over a finite field field:F5 linear representation theory of projective special linear group of degree two over a finite field
COMPARE AND CONTRAST: View linear representation theory of groups of order 60 to compare and contrast the linear representation theory with other groups of order 60.

## Degrees of irreducible representations

### Interpretation as alternating group

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

The partitions of 5 that are self-conjugate give irreducible representations of symmetric group:S5 that split into two irreducible representations of half the dimension each over alternating group:A5. Conjugate pairs of non-self-conjugate partitions of 5 restrict to equivalent irreducible representations over the alternating group.

Name(s) of representation(s) at symmetric groups level Partition or pair of partitions Self-conjugate case or conjugate pair case Degree(s) for representations of symmetric group Hook-length formula for degree Degree(s) of representations for alternating group
trivial, sign 5, 1 + 1 + 1 + 1 + 1 conjugate pair case 1, 1 $\frac{5!}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$ 1
standard, product of sign and standard 4 + 1, 2 + 1 + 1 + 1 conjugate pair case 4, 4 $\frac{5!}{5 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ 4
irreducible five-dimensional 3 + 2, 2 + 2 + 1 conjugate pair case 5, 5 $\frac{5!}{4 \cdot 3 \cdot 2 \cdot 1 \cdot 1}$ 5
exterior square of standard 3 + 1 + 1 self-conjugate 6 $\frac{5!}{5 \cdot 2 \cdot 2 \cdot 1 \cdot 1}$ 3, 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
Representation/conjugacy class representative and size $()$ (size 1) $(1,2)(3,4)$ (size 15) $(1,2,3)$ (size 20) $(1,2,3,4,5)$ (size 12) $(1,2,3,5,4)$ (size 12)
trivial 1 1 1 1 1
restriction of standard 4 0 1 -1 -1
irreducible five-dimensional 5 1 -1 0 0
one restriction of exterior square of standard 3 -1 0 $(\sqrt{5} - 1)/2$ $(-\sqrt{5} - 1)/2$
other restriction of exterior square of standard 3 -1 0 $(-\sqrt{5} - 1)/2$ $(\sqrt{5} - 1)/2$

## GAP implementation

The character degrees can be computed using GAP's CharacterDegrees function:

gap> CharacterDegrees(AlternatingGroup(5));
[ [ 1, 1 ], [ 3, 2 ], [ 4, 1 ], [ 5, 1 ] ]

This means that there is 1 degree 1 irreducible representation, 2 degree 3 irreducible representations, 1 degree 4 irreducible representation, and 1 degree 5 irreducible representation.

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

gap> Irr(CharacterTable(AlternatingGroup(5)));
[ Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 1, 1, 1, 1, 1 ] ),
Character( CharacterTable( Alt( [ 1 .. 5 ] ) ),
[ 3, -1, 0, -E(5)-E(5)^4, -E(5)^2-E(5)^3 ] ), Character( CharacterTable( Alt(
[ 1 .. 5 ] ) ), [ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ),
Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 4, 0, 1, -1, -1 ] ),
Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), [ 5, 1, -1, 0, 0 ] ) ]