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I am splitting off an entry classification of finite rotation groups from ADE classification in order to collect statements and references specific to the classification of finite subgroups of $SO(3)$ and $SU(2)$.
Is there a canonical reference for the proof of the classification statement? I find lots of lecture notes that give the proof, but all of them without citing sources or original publications of proofs.
I wrote:
Is there a canonical reference for the proof of the classification statement? I find lots of lecture notes that give the proof, but all of them without citing sources or original publications of proofs.
In Burban’s lecture notes (pdf) it has this statement:
The classification of finite isometry groups of $\mathbb{R}^3$ is a classical result of F. Klein (actually of Platon)
but still no concrete pointer…
according to Lamotke’s book, it seems that the result goes back to Klein 1884
(have not checked yet, need to run now)
Thanks for this! Excellent. I have added that reference also to all the entries on the separate finite rotation groups, as well as to that of the corresponding platonic solids.
Surely the binary cyclic groups (ie the even-order cyclic groups) are also pre-images of the cyclic group of half the order?
Yes, but not every cyclic subgroup of $SU(2)$ is in the image of taking pre-images. The odd order cycile groups are clearly not.
Oh, now I see. Thanks!
For what it’s worth I think the odd-order cyclic subgroups of $SU(2)$ are the image of the section $\mathbb{Z}/(2k+1) \to \mathbb{Z}/(4k+2) \simeq \mathbb{Z}/(2k+1) \times \mathbb{Z}/2$, so the odd-order cyclic subgroups of $SU(2)$ are index-2 subgroups of inverse images of finite subgroups of $SO(3)$, and the projection map $SU(2) \to SO(3)$ restricts to an isomorphism on these odd-order cyclic subgroups.
added statement of integral group (co-)homology of the finite subgroups of $SU(2)$ (here) together with a cross-pointer to discrete torsion of the sugra C-field
I am dabbling with drawing subgroup lattice under both $2O$ and $2I$ (a beginning here)
I didn’t fully appreciate before that $2T$ sits inside both of $2O$ and $2I$, forming a kind of axis of exceptional exceptionalism with $2D_4$. The three exceptional groups are not on the same footing, $2T$ is special.
Okay, I have produced a graphics of the full subgroup lattice of $SU(2)$ under the three exceptional finite subgroups. Now here.
It’s clearly not exactly symmetric under $2T \leftrightarrow 2T$ and $2O \leftrightarrow 2I$, but it comes surprisingly close. At least I had no idea that this is what the result was going to look like before I drew this.
This appendix seems to tally.
Given that $2 I$ is rather different with respect to normal subgroups (I guess for similar reasons that $A_5$ is simple), perhaps things are less similar than they seem.
But $2 D_4$ is a subgroup of $2 I$, so that needs an arrow.
[Edit: which of course is there indirectly.]
Thanks for the pdf. Can you see which publication this is the appendix of?
I should say that I am not really concerned about the symmetry-or-not about $2T$, but the fact that $2T$ takes a special place.
For instance $\beta$ is surjective over $\mathbb{R}$ for $2T$, but only surjective over $\mathbb{R}$ onto the integral characters for $2O$ and $2I$.
This has a curious consequence: If we do away with the irrational reps, then under McKay the corresponding vertices in the Dynkin diagram disappear, so the gauge group corresponding to the singularity should change. In other words, for $2O$ and $2I$, the McKay correspondence gets slightly modified as equivariant K-theory is replaced by equivariant stable cohomotopy, but not so for $2T$.
This may be relevant. GUT theory based on $2T \leftrightarrow E_6$ works. But GUT for $2O \leftrightarrow E_7$ and $2I \leftrightarrow E_8$ doesn’t actually work (at least not without bending over backwards), since these groups don’t admit chiral fermions.
It seems Cyclic Subgroups of the Sphere Braid Groups by Daciberg Lima Goncalves, John Guaschi, Springer.
Thanks! Had just found it, too, it’s on the arXiv: arXiv:1110.6628. Will add to the entry.
Just for the record, I found the association of reps to quiver vertices here (around p. 10 )
Turns out that
removing the two irrational nodes from the Dynkin diagram for 2O makes it become that of $2D_4$ and two disconnected nodes.
removing the four irrational nodes from the Dynkin diagram for 2I makes it become that of $A_4$ and one disconnected node.
now $A_4$ corresponds to $SU(5)$. So this case happens to match my little speculation… Not sure what to make of the other case, yet.
It’s KR-theory that’s at stake, isn’t it, rather than KO-theory?
A coincidence that it has been connected by Matthew Young with the representations of finite 2-groups (“This strongly suggests a role for Real 2-representation theory in M-theory”), as in the Platonic 2-groups for which Epa and Ganter remark: “The fact that there are canonical categorical extensions of all these groups suggests a categorical aspect of McKay correspondence that seems worth exploring” ?
We expect KR in general, but it reduces to KO at the orientifold fixed point.
Thanks for the pointer to Young, had not seen that.
From the maths ingredients it seems compelling to bring in the Platonic 2-groups. But despite some trying, I don’t see yet what’s really going on with them, in the physics story.
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