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I am wondering if there is a mistake in the real character table, taken from here.
Namely our computation shows that, $\rho_6 \oplus \rho_7$ (in my notation here) is a rational rep, hence $H_1/2 \oplus H_2/2$ in the notation there. But this would imply that this is also a real rep, which is in contradiction to the claim there that only $H_1 =2(H_1/2)$ and $H_2 =2(H_2/2)$ are real.
I’ve been in contact with James Montaldi. Will email him again…
Also he claims $Q_8$ is a normal subgroup, when this page says it isn’t. Who’s right?
Actually, maybe that page doesn’t say that. I took it from the table where there is no entry in the column
Size of each conjugacy class (=1 iff normal subgroup)
But then below it says it appears four times as a subgroup and once as a normal subgroup. Oh I see the empty entry was in the row
quaternion subgroup of one type
So there are different types.
OK, I’ll change that.
It should be possible to deduce the real reps from the complex character table at Groupprops here. The two rows which I suggest need to be added to get a real rep are precisely the two rows from the Groupprops page that involve the root of unity denoted there E(8)
.
I think the Groupnames page GL(2,3) agrees with my suggestion in #2 that the real irreps here are incorrect:
I gather that on Groupnames, as usual, they don’t state the Schur index precisely when it equals 1 – which here applies to all the complex irreps. But according to that other page it would 2 for the three reps in question.
I have sent a message to the GroupProps people, to confirm. But given that I have independent computation which is consistent with Schur index 1 for all irreps of 2O, I’ll go ahead now and change the character table on the $n$Lab accordingly.
Major relief: We found the resolution of the apparent contradiction regarding the real irrep characters of 2O.
Remember that the issue was that James Montaldi’s real character table for 2O here was in apparent conflict with the complex character table on the Groupnames page GL(2,3). Montaldi’s table implicitly has three Schur indices =2, while GL(2,3) has all Schur indices =1.
The resolution of this impasse is that “GroupNames” of all places, attached the wrong name to the group GL(2,3), and Wikipedia followed along. This is not 2O!! The binary octahedral group is instead CSU(2,3).
Now curiously GL(2,3) and CSU(2,3) have almost the same character tables… differing only in those Schur indices!
I just learned this from Tim Dokchitser, who I had asked for help with this. He says he has corrected both Wikipedia and Groupnames on this issue now, but that on Groupnames the correction will not become visible until some automatic updating mechanism will have picked it up.
And I will fix the $n$Lab entries now, accordingly.
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