[GAP Forum] Character Tables of Double Groups
Dmitrii Pasechnik
dmitrii.pasechnik at cs.ox.ac.uk
Tue May 27 13:09:18 BST 2014
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
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