[GAP Forum] Character Tables of Double Groups

[Au Eelis]

Dear Forum,

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

As I'm not really familiar with gap, I don't know, how to extract the names
of the irreducible representations and classes in this table, but I can see
that

X.1 -> A_1
X.2 -> A_2
X.5 -> E_1/2
X.6 -> E

1a -> E
2c -> -E
3a -> -2C_3
6a -> 2C_3

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.

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...

Does anyone know, what's wrong here?


Best regards,
Stefan