Linear representation theory of alternating group:A6

This article gives specific information, namely, linear representation theory, about a particular group, namely: alternating group:A6.
View linear representation theory of particular groups | View other specific information about alternating group:A6

Summary

Item	Value
Degrees of irreducible representations over a splitting field (such as $\mathbb {C}$ or ${\overline {\mathbb {Q} }}$ )	1,5,5,8,8,9,10 grouped form: 1 (1 time), 5 (2 times), 8 (2 times), 9 (1 time), 10 (1 time) maximum: 10, lcm: 360, number: 7, sum of squares: 360
Ring generated by character values	$\mathbb {Z} [2\cos(2\pi /5)]=\mathbb {Z} [(1+{\sqrt {5}})/2]$
Minimal splitting field, i.e., field of realization of all irreducible representations (characteristic zero)	$\mathbb {Q} ({\sqrt {5}})$ Quadratic extension of $\mathbb {Q}$ Same as field generated by character values
Orbits of irreducible representations under action of automorphism group	orbits of size 1 for representations of degree 1,9,10; orbits of size two for degree 5 and degree 8 representations (the degree 8 representations are interchanged under conjugation by an odd permutation; the degree 5 representations are interchaged by an automorphism that is outer for $S_{6}$ as well)
Orbits of irreducible representations under action of Galois group	orbits of size 1 for representations of degree 1,5,5,9,10; orbit of size two for degree 8 representations (automorphism ${\sqrt {5}}\mapsto -{\sqrt {5}}$ )
Minimal splitting field in prime characteristic $p\neq 2,3,5$	Case $p\equiv \pm 1{\pmod {5}}$ : prime field $\mathbb {F} _{p}$ Case $p\equiv \pm 2{\pmod {5}}$ : quadratic extension $\mathbb {F} _{p^{2}}$ of $\mathbb {F} _{p}$
Smallest size splitting field	field:F11
Degrees of irreducible representations over the rational numbers	1,5,5,9,10,16

Family contexts

Family name	Parameter values	General discussion of linear representation theory of family
alternating group	6	linear representation theory of alternating groups
projective special linear group of degree two over a finite field of size $q$	$q=9$ , i.e., field:F9, i.e., field of nine elements, so the group is $PSL(2,9)$	linear representation theory of projective special linear group of degree two over a finite field

GAP implementation

The degrees of irreducible representations can be found using GAP's CharacterDegrees and AlternatingGroup functions:

gap> CharacterDegrees(AlternatingGroup(6));
[ [ 1, 1 ], [ 5, 2 ], [ 8, 2 ], [ 9, 1 ], [ 10, 1 ] ]

This means that there is 1 degree 1 irreducible, 2 degree 5 irreducibles, 2 degree 8 irreducibles, 1 degree 9 irreducible, and 1 degree 10 irreducible representation.

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

gap> Irr(CharacterTable(AlternatingGroup(6)));
[ Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 1, 1, 1, 1, 1, 1, 1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 5, 1, 2, -1, -1, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 5, 1, -1, 2, -1, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 8, 0, -1, -1, 0, -E(5)-E(5)^4,
      -E(5)^2-E(5)^3 ] ), Character( CharacterTable( Alt( [ 1 .. 6 ] ) ),
    [ 8, 0, -1, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] ),
  Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 9, 1, 0, 0, 1, -1, -1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 6 ] ) ), [ 10, -2, 1, 1, 0, 0, 0 ] ) ]