Linear representation theory of binary octahedral group

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

The binary octahedral group is a binary von Dyck group with parameters $(4,3,2)$, i.e., it has the presentation:

$\langle a,b,c \mid a^4 = b^3 = c^2 = abc\rangle$.

We denote the element $a^4 = b^3 = c^2$ as $z$. This element has order two.

This article discusses the linear representation theory of the binary octahedral group in characteristics other than 2 and 3.

Summary

Item	Value
degrees of irreducible representations over a splitting field	1,1,2,2,2,3,3,4 maximum: 4, lcm: 12, number: 8, sum of squares: 48
ring generated by character values (characteristic zero)	$\mathbb{Z}[\sqrt{2}]$, same as $\mathbb{Z}[t]/(t^2 - 2)$
field generated by character values (characteristic zero)	$\mathbb{Q}(\sqrt{2})$, same as $\mathbb{Q}[t]/(t^2 - 2)$

Note that general linear group:GL(2,3) and the binary octahedral group are isoclinic groups of the same order. We know that isoclinic groups have same proportions of degrees of irreducible representations, therefore, in this case, the degrees of irreducible representations are the same for both groups. However, the character tables themselves are not identical. In fact, the fields generated by character values also differ from one another.

GAP implementation

Degrees of irreducible representations

The degrees of irreducible representations can be computed using GAP's CharacterDegrees function:

gap> CharacterDegrees(SmallGroup(48,28));
[ [ 1, 2 ], [ 2, 3 ], [ 3, 2 ], [ 4, 1 ] ]

Character table

The character table can be computed using GAP's CharacterTable function:

gap> Irr(CharacterTable(SmallGroup(48,28)));
[ Character( CharacterTable( <pc group of size 48 with 5 generators> ),
    [ 1, 1, 1, 1, 1, 1, 1, 1 ] ),
  Character( CharacterTable( <pc group of size 48 with 5 generators> ),
    [ 1, -1, 1, 1, 1, -1, 1, -1 ] ),
  Character( CharacterTable( <pc group of size 48 with 5 generators> ),
    [ 2, 0, -1, 2, 2, 0, -1, 0 ] ),
  Character( CharacterTable( <pc group of size 48 with 5 generators> ),
    [ 2, 0, -1, 0, -2, -E(8)+E(8)^3, 1, E(8)-E(8)^3 ] ),
  Character( CharacterTable( <pc group of size 48 with 5 generators> ),
    [ 2, 0, -1, 0, -2, E(8)-E(8)^3, 1, -E(8)+E(8)^3 ] ),
  Character( CharacterTable( <pc group of size 48 with 5 generators> ),
    [ 3, 1, 0, -1, 3, -1, 0, -1 ] ),
  Character( CharacterTable( <pc group of size 48 with 5 generators> ),
    [ 3, -1, 0, -1, 3, 1, 0, 1 ] ),
  Character( CharacterTable( <pc group of size 48 with 5 generators> ),
    [ 4, 0, 1, 0, -4, 0, -1, 0 ] ) ]

The character table can be displayed more elegantly using the Display function:

gap> Display(CharacterTable(SmallGroup(48,28)));
CT1

     2  4  2  1  3  4  3  1  3
     3  1  .  1  .  1  .  1  .

       1a 4a 3a 4b 2a 8a 6a 8b
    2P 1a 2a 3a 2a 1a 4b 3a 4b
    3P 1a 4a 1a 4b 2a 8b 2a 8a
    5P 1a 4a 3a 4b 2a 8b 6a 8a
    7P 1a 4a 3a 4b 2a 8a 6a 8b

X.1     1  1  1  1  1  1  1  1
X.2     1 -1  1  1  1 -1  1 -1
X.3     2  . -1  2  2  . -1  .
X.4     2  . -1  . -2  A  1 -A
X.5     2  . -1  . -2 -A  1  A
X.6     3  1  . -1  3 -1  . -1
X.7     3 -1  . -1  3  1  .  1
X.8     4  .  1  . -4  . -1  .

A = -E(8)+E(8)^3
  = -Sqrt(2) = -r2

Irreducible representations

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

gap> IrreducibleRepresentations(SmallGroup(48,28));
[ [ f1, f2, f3, f4, f5 ] -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ],
      [ [ 1 ] ] ],
  [ f1, f2, f3, f4, f5 ] -> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ],
      [ [ 1 ] ] ],
  [ f1, f2, f3, f4, f5 ] ->
    [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ -1, -1 ], [ 1, 0 ] ],
      [ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 0, 1 ] ]
     ],
  [ f1, f2, f3, f4, f5 ] ->
    [
      [ [ -1/2*E(8)+E(8)^2+1/2*E(8)^3, -1+1/2*E(8)+1/2*E(8)^3 ],
          [ 1-1/2*E(8)-1/2*E(8)^3, 1/2*E(8)-E(8)^2-1/2*E(8)^3 ] ],
      [ [ -1-1/2*E(8)-1/2*E(8)^3, -1/2*E(8)-E(8)^2+1/2*E(8)^3 ],
          [ 1/2*E(8)-1/2*E(8)^3, 1/2*E(8)+1/2*E(8)^3 ] ],
      [ [ -1, -E(8)+E(8)^3 ], [ E(8)-E(8)^3, 1 ] ],
      [ [ E(8)+E(8)^3, E(4) ], [ -E(4), -E(8)-E(8)^3 ] ],
      [ [ -1, 0 ], [ 0, -1 ] ] ],
  [ f1, f2, f3, f4, f5 ] ->
    [
      [ [ -1/2*E(8)+E(8)^2+1/2*E(8)^3, 1-1/2*E(8)-1/2*E(8)^3 ],
          [ -1+1/2*E(8)+1/2*E(8)^3, 1/2*E(8)-E(8)^2-1/2*E(8)^3 ] ],
      [ [ 1/2*E(8)+1/2*E(8)^3, -1/2*E(8)+1/2*E(8)^3 ],
          [ 1/2*E(8)+E(8)^2-1/2*E(8)^3, -1-1/2*E(8)-1/2*E(8)^3 ] ],
      [ [ 1, -E(8)+E(8)^3 ], [ E(8)-E(8)^3, -1 ] ],
      [ [ -E(8)-E(8)^3, E(4) ], [ -E(4), E(8)+E(8)^3 ] ],
      [ [ -1, 0 ], [ 0, -1 ] ] ],
  [ f1, f2, f3, f4, f5 ] -> [ [ [ 1, 1, 0 ], [ 0, -1, 0 ], [ 0, -1, 1 ] ],
      [ [ 1, 1, 0 ], [ 0, -1, 1 ], [ 0, -1, 0 ] ],
      [ [ -1, 0, 0 ], [ 1, 0, 1 ], [ 1, 1, 0 ] ],
      [ [ 0, -1, 1 ], [ -1, 0, -1 ], [ 0, 0, -1 ] ],
      [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ],
  [ f1, f2, f3, f4, f5 ] -> [ [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 1, 1, -1 ] ],
      [ [ 0, -1, 1 ], [ -1, 0, 0 ], [ -1, -1, 0 ] ],
      [ [ 0, 0, 1 ], [ 0, -1, 0 ], [ 1, 0, 0 ] ],
      [ [ -1, 0, 0 ], [ 1, 1, -1 ], [ 0, 0, -1 ] ],
      [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ],
  [ f1, f2, f3, f4, f5 ] ->
    [ [ [ 0, 0, -E(3)^2, 1 ], [ 0, 0, 1, E(3)^2 ], [ -1, E(3), 0, 0 ],
          [ E(3), 1, 0, 0 ] ],
      [ [ E(3), 0, 0, 0 ], [ -E(3)^2, 1, 0, 0 ], [ 0, 0, -E(3), -1 ],
          [ 0, 0, E(3)^2, 0 ] ],
      [ [ 0, 1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, E(3), -E(3)^2 ],
          [ 0, 0, -E(3)^2, -E(3) ] ],
      [ [ E(3), -E(3)^2, 0, 0 ], [ -E(3)^2, -E(3), 0, 0 ], [ 0, 0, 0, -1 ],
          [ 0, 0, 1, 0 ] ],
      [ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ]
     ] ]