# 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

## Back story

This page gives information about the degrees of irreducible representations, character table, and irreducible linear representations of alternating group:A5. It does not, however, provide an adequate explanation of how one might arrive at (i.e., deduce) the information. For more on the back story, see determination of character table of alternating group:A5.

## 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 ] ] ] ]