Linear representation theory of alternating group:A9

This article gives specific information, namely, linear representation theory, about a particular group, namely: alternating group:A9.
View linear representation theory of particular groups | View other specific information about alternating group:A9

Summary

Item	Value
degrees of irreducible representations over a splitting field (such as ${\overline {\mathbb {Q} }}$ or $\mathbb {C}$)	1, 8, 21, 21, 27, 28, 35, 35, 42, 48, 56, 84, 105, 120, 162, 168, 189, 216 grouped form (each occurs once by default): 1, 8, 21 (2 times), 27, 28, 35 (2 times), 42, 48, 56, 84, 105, 120, 162, 168, 189, 216 maximum: 216, number: 18, sum of squares: 181440
minimal splitting field, i.e., smallest field of realization of all irreducible representations (characteristic zero)	$\mathbb {Q} (\zeta +\zeta ^{2}+\zeta ^{4}+\zeta ^{8})$ where $\zeta$ is a primitive fifteenth root of unity Same as $\mathbb {Q} ({\sqrt {-15}})$ Same as field generated by character values
condition for a field of characteristic not 2,3,5,7 to be a splitting field	-15 should be a square in that field

GAP implementation

Degrees of irreducible representations

These can be computed using the CharacterDegrees and CharacterTable functions:

gap> CharacterDegrees(CharacterTable(AlternatingGroup(9)));
[ [ 1, 1 ], [ 8, 1 ], [ 21, 2 ], [ 27, 1 ], [ 28, 1 ], [ 35, 2 ], [ 42, 1 ], [ 48, 1 ], [ 56, 1 ], [ 84, 1 ], [ 105, 1 ], [ 120, 1 ], [ 162, 1 ],
  [ 168, 1 ], [ 189, 1 ], [ 216, 1 ] ]

Character table

These can be computed using the Irr and CharacterTable functions:

gap> Irr(CharacterTable(AlternatingGroup(9)));
[ Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 8, 4, 0, 5, 1, 2, -1, 2, -1, 0, 3, -1, 0, 0, 0, 1, -1, -1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 21, 1, -3, -3, 1, 0, 3, -1, -1, 1, 1, 1,
      -E(15)-E(15)^2-E(15)^4-E(15)^8, -E(15)^7-E(15)^11-E(15)^13-E(15)^14,
      0, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 21, 1, -3, -3, 1, 0, 3, -1, -1, 1, 1, 1,
      -E(15)^7-E(15)^11-E(15)^13-E(15)^14, -E(15)-E(15)^2-E(15)^4-E(15)^8,
      0, 0, 0, 0 ] ), Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 27, 7, 3, 9, 1, 0, 0, 1, 1, -1, 2, 2, -1, -1, 0, -1, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 28, 4, -4, 10, -2, 1, 1, 0, 0, 0, 3, -1, 0, 0, -1, 0, 1, 1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 35, -5, 3, 5, 1, 2, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, -1, 2 ] ),
  Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 35, -5, 3, 5, 1, 2, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 2, -1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 42, 6, 2, 0, 0, 3, -3, 0, 0, 2, -3, 1, 0, 0, -1, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 48, 8, 0, 6, 2, 0, 3, 0, 0, 0, -2, -2, 1, 1, 0, -1, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 56, -4, 0, 11, -1, 2, 2, -2, 1, 0, 1, 1, 1, 1, 0, 0, -1, -1 ] ),
  Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 84, 4, 4, -6, -2, 3, 3, 0, 0, 0, -1, -1, -1, -1, 1, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 105, 5, 1, 15, -1, -3, -3, -1, -1, 1, 0, 0, 0, 0, 1, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 120, 0, 8, 0, 0, -3, 3, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 162, 6, -6, 0, 0, 0, 0, 0, 0, -2, -3, 1, 0, 0, 0, 1, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 168, 4, 0, -15, 1, 0, -3, -2, 1, 0, 3, -1, 0, 0, 0, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 189, -11, -3, 9, 1, 0, 0, 1, 1, 1, -1, -1, -1, -1, 0, 0, 0, 0 ] ),
  Character( CharacterTable( Alt( [ 1 .. 9 ] ) ),
    [ 216, -4, 0, -9, -1, 0, 0, 2, -1, 0, 1, 1, 1, 1, 0, -1, 0, 0 ] ) ]