[GAP Forum] Character Tables of Double Groups

[Dmitrii Pasechnik]

On Tue, May 27, 2014 at 12:56:54PM +0200, Au Eelis wrote:
> my current field of work is the analysis of electronic band structures,
> which were calculated with spin-orbit coupling. To analyse such band
> structures, you need the double space groups and their character tables, to
> get information about the irreducible representations which transform like
> the corresponding bands.
> 
> Unfortunately, I have difficulties, to get character tables, which
> correspond to the literature. An easy example would be the character table
> of the double group of C3v. In literature you find this character table
> very often and it looks like this:
> 
>        | E     | 2C_3  | 3s_v  | -E    | -2C_3 | -3s_v |
> -------+-------+-------+-------+-------+-------+-------+
> A_1    |     1 |     1 |     1 |     1 |     1 |     1 |
> A_2    |     1 |     1 |    -1 |     1 |     1 |    -1 |
> E      |     2 |    -1 |     0 |     2 |    -1 |     0 |
> E_1/2  |     2 |     1 |     0 |    -2 |    -1 |     0 |
> 1E_3/2 |     1 |    -1 |     i |    -1 |     1 |    -i |
> 2E_3/2 |     1 |    -1 |    -i |    -1 |     1 |     i |
> 
> Now I wanted to reproduce this character table with gap. At first, I create
> this group with gap using the threefold rotation around z and the
> reflection at the x-axis as generators (in representation U(2)):
> 
> gap> rep:=[
> > [[1/2, -Sqrt(-3)/2],[-Sqrt(-3)/2, 1/2]],
> > [[1, 0],[0, -1]]
> > ];
> gap> h:=Group(rep);
> gap> Display(CharacterTable(h));
> 
>      2  2  2  1  2  1  2
>      3  1  .  1  .  1  1
> 
>        1a 2a 3a 2b 6a 2c
> 
> X.1     1  1  1  1  1  1
> X.2     1 -1  1 -1  1  1
> X.3     1  1  1 -1 -1 -1
> X.4     1 -1  1  1 -1 -1
> X.5     2  . -1  .  1 -2
> X.6     2  . -1  . -1  2
[...]
> The big problem can be seen in the irreducible representations X.3/X.4 (or
> 1E_3/2 and 2E_3/2), where the literature predicts complex characters, while
> GAP shows non-complex values.

You constructed above a different group.
The group you constructed is
isomorphic to the dihedral group of order 12, i.e. the group of
symmetries of the 6-gon.

But the table you gave is from a different group of order 12, which does have 
cyclic Sylow 2-subgroups.
We can browse the character tables of the 5 order 12 groups in GAP, 
as follows: 

for k in [1..5] do Display(CharacterTable(SmallGroup(12,k))); od;

The group with the character table as you gave above is 
the one for k=1. It can be constructed as a matrix group as
as follows:
gg:=Group([ [ [ 0, -1 ], [ 1, 0 ] ], 
[ [ -1, 0 ], [ 0, -1 ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ] ]]);

Now we look at its character table (in GAP):
gap> Display(CharacterTable(gg));
CT15

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

       1a 4a 4b 2a 6a 3a

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

A = -E(4)
  = -Sqrt(-1) = -i

> 
> At the moment, I don't know, where to look for the problem. My possible
> thoughts are: wrong generators, problem with the algorithms in GAP or wrong
> literature...

wrong generators in your case...

HTH,
Dmitrii