# 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 (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$) 1,3,3,4,5
grouped form: 1 (1 time), 3 (2 times), 4 (1 time), 5 (1 time)
maximum: 5, lcm: 60, number: 5, sum of squares: 60, quasirandom degree: 3
Schur index values of irreducible representations 1,1,1,1,1
Ring generated by character values $\mathbb{Z}[(1 + \sqrt{5})/2]$ or $\mathbb{Z}[2\cos(2\pi/5)]$
Minimal splitting field, i.e., smallest field of realization for all irreducible representations $\mathbb{Q}(\sqrt{5})$ -- quadratic extension of field of rational numbers
Same as field generated by character values
Orbit structure under action of automorphism group orbits of size 1 each of degree 1, 4, and 5 representations, orbit of size 2 of degree 3 representations (interchanged via an automorphism induced by conjugation by odd permutation)
Orbit structure under action of Galois group over rationals orbits of size 1 each of degree 1, 4, and 5 representations, orbit of size 2 of degree 3 representations (interchanged via the mapping $\sqrt{5} \mapsto -\sqrt{5}$)
Degrees of irreducible representations over the field of rational numbers 1,4,5,6

## 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 of size $q$ $q = 4$, i.e., field:F4, so the group is $PGL(2,4)$ 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 of size $q$ $q = 5$, i.e., field:F5 , so the group is $PSL(2,5)$ 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 irreducible constituent of restriction of exterior square of standard 3 -1 0 $(\sqrt{5} +1)/2$ $(-\sqrt{5} + 1)/2$
other irreducible constituent of restriction of exterior square of standard 3 -1 0 $(-\sqrt{5} + 1)/2$ $(\sqrt{5} + 1)/2$

Below are the size-degree-weighted characters, obtained by multiplying each 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) $(1,2)(3,4)$ (size 15) $(1,2,3)$ (size 20) $(1,2,3,4,5)$ (size 12) $(1,2,3,5,4)$ (size 12)
trivial 1 15 20 12 12
restriction of standard 1 0 5 -3 -3
irreducible five-dimensional 1 3 -4 0 0
one restriction of exterior square of standard 1 -5 0 $2(\sqrt{5} + 1)$ $2(-\sqrt{5} + 1)$
other restriction of exterior square of standard 1 -5 0 $2(-\sqrt{5} + 1)$ $2(\sqrt{5} + 1)$

## GAP implementation

### Degrees of irreducible representations

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.

### Character table

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 ] ) ]

The character table can be displayed somewhat more nicely as follows:

gap> Display(CharacterTable(AlternatingGroup(5)));
CT2

2  2  2  .  .  .
3  1  .  1  .  .
5  1  .  .  1  1

1a 2a 3a 5a 5b
2P 1a 1a 3a 5b 5a
3P 1a 2a 1a 5b 5a
5P 1a 2a 3a 1a 1a

X.1     1  1  1  1  1
X.2     3 -1  .  A *A
X.3     3 -1  . *A  A
X.4     4  .  1 -1 -1
X.5     5  1 -1  .  .

A = -E(5)-E(5)^4
= (1-ER(5))/2 = -b5

### Irreducible representations

The irreducible representations can be computed using GAP's IrreducibleRepresentations function as follows:

gap> IrreducibleRepresentations(AlternatingGroup(5));
[ [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ],
[ (1,2,3,4,5), (3,4,5) ] ->
[ [ [ 0, -1, 0 ], [ -E(5)-E(5)^4, -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4, 1 ],
[ 2*E(5)+E(5)^2+E(5)^3+2*E(5)^4, 2*E(5)+E(5)^2+E(5)^3+2*E(5)^4, -1 ]
], [ [ 1, -E(5)-E(5)^4, -E(5)-E(5)^4 ], [ 0, -1, -1 ], [ 0, 1, 0 ] ] ]
,
[ (1,2,3,4,5), (3,4,5) ] -> [ [ [ 1, 0, 1 ], [ -1, 0, 0 ], [ -1, -1, E(5)+E(5\
)^4 ] ], [ [ 0, -E(5)^2-E(5)^3, E(5)^2+E(5)^3 ], [ -1, -1, E(5)+E(5)^4 ],
[ -E(5)-E(5)^4, -E(5)-E(5)^4, 1 ] ] ],
[ (1,2,3,4,5), (3,4,5) ] -> [ [
[ E(3)^2, -1/3*E(3)-2/3*E(3)^2, E(3)^2, 4/3*E(3)+2/3*E(3)^2 ],
[ E(3), 4/3*E(3)+2/3*E(3)^2, 1, -1/3*E(3)-2/3*E(3)^2 ],
[ E(3), E(3), 0, -E(3)^2 ],
[ E(3), -2/3*E(3)-1/3*E(3)^2, E(3), -1/3*E(3)-2/3*E(3)^2 ] ],
[ [ E(3), 1/3*E(3)-1/3*E(3)^2, 0, 2/3*E(3)-2/3*E(3)^2 ],
[ E(3)^2, 0, 0, 0 ],
[ 0, 2/3*E(3)+1/3*E(3)^2, 1, -2/3*E(3)-1/3*E(3)^2 ],
[ 1, 1, 0, -E(3) ] ] ],
[ (1,2,3,4,5), (3,4,5) ] -> [ [ [ -1, -1, -1, -1, -1 ], [ 1, 1, 0, 0, 1 ],
[ 0, -1, -1, 0, -1 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 1, 1 ] ],
[ [ -1, -1, -1, -1, -1 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1 ],
[ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ] ] ] ]