Linear representation theory of alternating group:A5

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This article gives specific information, namely, linear representation theory, about a particular group, namely: 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

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

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 ] ) ]