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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 15th 2018
    • (edited Apr 15th 2018)

    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)SO(3) and SU(2)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.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 15th 2018

    found a good textbook account: Rees 05

    but again there is no pointer in there to the origin of the statement.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 15th 2018

    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 ℝ 3\mathbb{R}^3 is a classical result of F. Klein (actually of Platon)

    but still no concrete pointer…

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 15th 2018

    according to Lamotke’s book, it seems that the result goes back to Klein 1884

    (have not checked yet, need to run now)

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 15th 2018

    From here

    In his Vorlesungen über das Ikosaeder [Klein,1993], published in 1884, Felix Klein gives the classification of finite subgroups of SL(2,ℂ)SL(2,\mathbb{C}) up to conjugacy

    There’s an online translation here.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 15th 2018

    I added bibliographic detail about Klein’s work and the translation.

    diff, v4, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2018

    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.

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 17th 2018

    Surely the binary cyclic groups (ie the even-order cyclic groups) are also pre-images of the cyclic group of half the order?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 17th 2018

    Yes, but not every cyclic subgroup of SU(2)SU(2) is in the image of taking pre-images. The odd order cycile groups are clearly not.

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 17th 2018

    Tweaked description of which finite subgroups are preimages under the double cover projection.

    diff, v9, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 17th 2018

    Oh, now I see. Thanks!

    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 17th 2018

    For what it’s worth I think the odd-order cyclic subgroups of SU(2)SU(2) are the image of the section ℤ/(2k+1)→ℤ/(4k+2)≃ℤ/(2k+1)×ℤ/2\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)SU(2) are index-2 subgroups of inverse images of finite subgroups of SO(3)SO(3), and the projection map SU(2)→SO(3)SU(2) \to SO(3) restricts to an isomorphism on these odd-order cyclic subgroups.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2018

    added statement of integral group (co-)homology of the finite subgroups of SU(2)SU(2) (here) together with a cross-pointer to discrete torsion of the sugra C-field

    diff, v12, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2018

    adjusted page title

    diff, v12, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeOct 7th 2018
    • (edited Oct 7th 2018)

    I am dabbling with drawing subgroup lattice under both 2O2O and 2I2I (a beginning here)

    I didn’t fully appreciate before that 2T2T sits inside both of 2O2O and 2I2I, forming a kind of axis of exceptional exceptionalism with 2D 42D_4. The three exceptional groups are not on the same footing, 2T2T is special.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeOct 8th 2018
    • (edited Oct 8th 2018)

    Okay, I have produced a graphics of the full subgroup lattice of SU(2)SU(2) under the three exceptional finite subgroups. Now here.

    It’s clearly not exactly symmetric under 2T↔2T2T \leftrightarrow 2T and 2O↔2I2O \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.

    • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 8th 2018

    This appendix seems to tally.

    Given that 2I2 I is rather different with respect to normal subgroups (I guess for similar reasons that A 5A_5 is simple), perhaps things are less similar than they seem.

    • CommentRowNumber18.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 8th 2018
    • (edited Oct 8th 2018)

    But 2D 42 D_4 is a subgroup of 2I2 I, so that needs an arrow.

    [Edit: which of course is there indirectly.]

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeOct 8th 2018

    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 2T2T, but the fact that 2T2T takes a special place.

    For instance β\beta is surjective over ℝ\mathbb{R} for 2T2T, but only surjective over ℝ\mathbb{R} onto the integral characters for 2O2O and 2I2I.

    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 2O2O and 2I2I, the McKay correspondence gets slightly modified as equivariant K-theory is replaced by equivariant stable cohomotopy, but not so for 2T2T.

    This may be relevant. GUT theory based on 2T↔E 62T \leftrightarrow E_6 works. But GUT for 2O↔E 72O \leftrightarrow E_7 and 2I↔E 82I \leftrightarrow E_8 doesn’t actually work (at least not without bending over backwards), since these groups don’t admit chiral fermions.

    • CommentRowNumber20.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 8th 2018

    It seems Cyclic Subgroups of the Sphere Braid Groups by Daciberg Lima Goncalves, John Guaschi, Springer.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeOct 8th 2018

    Thanks! Had just found it, too, it’s on the arXiv: arXiv:1110.6628. Will add to the entry.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeOct 8th 2018
    • (edited Oct 8th 2018)

    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 42D_4 and two disconnected nodes.

    • removing the four irrational nodes from the Dynkin diagram for 2I makes it become that of A 4A_4 and one disconnected node.

    now A 4A_4 corresponds to SU(5)SU(5). So this case happens to match my little speculation… Not sure what to make of the other case, yet.

    • CommentRowNumber23.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 8th 2018

    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” ?

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2018

    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.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2018

    added pointer to

    • Mattia Mecchia, Bruno Zimmermann, On finite groups acting on homology 4-spheres and finite subgroups of SO(5)SO(5), Topology and its Applications 158.6 (2011): 741-747 (arXiv:1001.3976)

    for classification of the finite subgroups of O(5)O(5).

    diff, v21, current

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2018

    for classification of finite subgroups of O(4)O(4) I have added pointer to

    • Patrick du Val, Homographies, Quaternions and Rotations, Oxford Mathematical Monographs, Clarendon Press (1964)

      also(?): Journal of the London Mathematical Society, Volume s1-40, Issue 1 (1965) (doi:10.1112/jlms/s1-40.1.569b)

    diff, v21, current

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2018
    • (edited Dec 7th 2018)

    for finite subgroups of O(4)O(4) also:

    • John Conway, D. A. Smith, On quaternions and octonions: their geometry, arithmetic and symmetry A K Peters Ltd., Natick, MA, 2003

    • Paul de Medeiros, José Figueroa-O’Farrill, appendix B of Half-BPS M2-brane orbifolds, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (arXiv:1007.4761, Euclid)

    Which of these finite subgroups is the symmetry of the 120-cell?

    diff, v21, current

    • CommentRowNumber28.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 7th 2018
    • (edited Dec 7th 2018)

    The exceptional Coxeter group H 4H_4 according to Dechant here, and Wikipedia agrees.

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2018

    Thanks. Can you figure out which group that is in terms of the list of all subgroups in tables 16, 17 and 18 in arXiv:1007.4761 ? (Beware that these tables are in the middle of the References, scattered over the very last pages).

    • CommentRowNumber30.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 7th 2018
    • (edited Dec 7th 2018)

    This Google books page claims that the orientation preserving symmetries of the 120-cell (and so 600-cell) is Y′×Y′/ℤ 2Y' \times \Y'/\mathbb{Z}_2, where Y′Y' is the binary icosohedral group. So that makes it the final entry in one of those tables.

    • CommentRowNumber31.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 7th 2018

    So ±[I×I]\pm [I\times I].

    • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2018

    Thanks a lot! Have added it here.

    Might you also know the corresponding identification for the symmetry group of the 24-cell?

    diff, v22, current

    • CommentRowNumber33.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 7th 2018

    So the full group is the Coxeter group F 4F_4 with 1152 elements. Then the orientation-preserving subgroup of SO(4)SO(4) is of order 576. But how does it break down relative to the double cover Spin(4)≡Sp(1)×Sp(1)→SO(4)Spin(4) \equiv Sp(1) \times Sp(1) \to SO(4)?

    • CommentRowNumber34.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 7th 2018

    What’s this telling us:

    There is an analogy in four dimensions, where the 24-cell and its dual can be composed to yield an object with a rotational symmetry group of order 1152 instead of 576. The double cover of this, with order 2304, is obtained by choosing any ordered pair (l,r) of quaternions from the binary octahedral group 2O.

    Recall that 2O has 2T as a normal subgroup, allowing us to refer meaningfully to ‘odd’ and ‘even’ rotations. If we add the additional constraint that l and r have the same parity, we get down to a group of order 1152, the double cover of the symmetry group of the 24-cell.

    Does that make the right subgroup of SO(4)SO(4) isomorphic to ±12[O×O]\pm \frac{1}{2}[O \times O]?

    • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2018

    Thanks again! Much appreciated. Sounds plausible. Would need to think about this.

    The phrase “24-cell and its dual” sounds a little surprising, given that the 24-cell is supposed to be self-dual. But I guess the group action may depend on whether we think of it one way or the other?

    • CommentRowNumber36.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 7th 2018

    Presumably that would be like taking a tetrahedron and then superimposing it on its (isomorphic) dual as in Compound of two tetrahedra.

    • CommentRowNumber37.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2018

    Right, that must be it.

    • CommentRowNumber38.
    • CommentAuthorUrs
    • CommentTimeFeb 18th 2019
    • (edited Feb 18th 2019)

    in making up for my sins in another thread, I have added more detail on how exactly all the finite subgroups of SU(2)SU(2) sit inside and how they map to SO(3)SO(3). Now it reads like so:

    (…)

    Here under the double cover projection (using the exceptional isomorphism SU(2)≃Spin(3)SU(2) \simeq Spin(3))

    SU(2)≃Spin(3)⟶πSO(3) SU(2) \simeq Spin(3) \overset{\pi}{\longrightarrow} SO(3)

    all the finite subgroups of SU(2)SU(2) except the odd-order cyclic groups are the preimages of the corresponding finite subgroups of SO(3)SO(3), in that we have pullback diagrams

    ⟨exp(πi1n)⟩ = ℤ/(2n) ↪AA Spin(2) ↪AA Spin(3) ↓ (pb) ↓ (pb) ↓ π ⟨Ad exp(πi1n)⟩ = ℤ/n ↪AA SO(2) ↪AA SO(3) \array{ \left\langle \exp \left( \pi \mathrm{i} \tfrac{1}{n} \right) \right\rangle & = & \mathbb{Z}/(2n) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ && \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{ \exp \left( \pi \mathrm{i} \tfrac{1}{n} \right) } \right\rangle & = & \mathbb{Z}/n &\overset{\phantom{AA}}{\hookrightarrow}& SO(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }

    exhibiting the even order cyclic groups as subgroups of Spin(2), including the the minimal case of the group of order 2

    ⟨exp(πi)=−1⟩ = ℤ/2 ↪AA Spin(2) ↪AA Spin(3) ↓ (pb) ↓ (pb) ↓ π ⟨Ad exp(πi)=e⟩ = 1 ↪AA SO(2) ↪AA SO(3) \array{ \left\langle \exp \left( \pi \mathrm{i} \right) = -1 \right\rangle & = & \mathbb{Z}/2 &\overset{\phantom{AA}}{\hookrightarrow}& Spin(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ && \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{ \exp \left( \pi \mathrm{i} \right) } = e \right\rangle & = & 1 &\overset{\phantom{AA}}{\hookrightarrow}& SO(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }

    as well as

    ⟨exp(πi1n),j⟩ = 2D 2n ↪AA Pin −(2) ↪AA Spin(3) ↓ (pb) ↓ (pb) ↓ π ⟨Ad exp(πi1n),Ad j⟩ D 2n ↪AA O(2) ↪AA SO(3) \array{ \left\langle \exp\left( \pi \mathrm{i} \tfrac{1}{n} \right), \, \mathrm{j} \right\rangle &=& 2 D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& Pin_-(2) &\overset{\phantom{AA}}{\hookrightarrow}& Spin(3) \\ && \big\downarrow &{}^{(pb)}& \big\downarrow &{}^{(pb)}& \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{\exp\left( \pi \mathrm{i} \tfrac{1}{n} \right) }, \, Ad_{\mathrm{j}} \right\rangle && D_{2n} &\overset{\phantom{AA}}{\hookrightarrow}& O(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }

    exhibiting the binary dihedral groups as sitting inside the Pin(2)-subgroup of Spin(3),

    but only commuting diagrams

    ⟨exp(2πi12n+1)⟩ = ℤ/(2n+1) ↪AA Spin(3) ↓ ↓ π ⟨Ad exp(2πi12n+1)⟩ = ℤ/(2n+1) ↪AA SO(2) ↪AA SO(3) \array{ \left\langle \exp \left( 2 \pi \mathrm{i} \tfrac{1}{{2n+1}} \right) \right\rangle & = & \mathbb{Z}/(2n+1) &&\overset{\phantom{AA}}{\hookrightarrow}&& Spin(3) \\ && \big\downarrow && && \big\downarrow^{ \mathrlap{\pi} } \\ \left\langle Ad_{ \exp \left( 2 \pi \mathrm{i} \tfrac{1}{2n+1} \right) } \right\rangle & = & \mathbb{Z}/(2n+1) &\overset{\phantom{AA}}{\hookrightarrow}& SO(2) &\overset{\phantom{AA}}{\hookrightarrow}& SO(3) }

    for the odd order cyclic subgroups.

    diff, v24, current

    • CommentRowNumber39.
    • CommentAuthorUrs
    • CommentTimeApr 11th 2019
    • (edited Apr 11th 2019)

    [ removed, sorry for the noise ]

    • CommentRowNumber40.
    • CommentAuthorUrs
    • CommentTimeOct 5th 2019
    • (edited Oct 5th 2019)

    The entry states the group cohomology of finite subgroups of SU(2)SU(2) as an abelian group:

    H grp n(G ADE,ℤ)≃{ℤ | n=0 G ADE ab | n=2mod4 ℤ/|G ADE| | npositive multiple of4 0 | otherwise H^n_{grp}(G_{ADE}, \mathbb{Z}) \;\simeq\; \left\{ \array{ \mathbb{Z} &\vert& n = 0 \\ G_{ADE}^{ab} &\vert& n = 2 \, mod \, 4 \\ \mathbb{Z}/{\vert G_{ADE}\vert} &\vert& n \, \text{positive multiple of} \, 4 \\ 0 &\vert& \text{otherwise} } \right.

    Question: What’s the ring structure under cup product?

    In particular, what’s

    H 2⊗H 2⟶∪H 4 H^2 \otimes H^2 \overset{\cup}{\longrightarrow} H^4

    ??

    Partial answers:

    For cyclic groups it’s clear.

    For generalized quaternion groups ( in fact for all binary dihedral groups) I see it’s Theorem 3 in

    • Takao Hayami, Katsunori Sanada: Cohomology ring of the generalized quaternion group with coefficients in an order, Communications in Algebra 30(8):3611-3628 (2002)  doi:10.1081/AGB-120005809

    For the ordinary quaternion group and for the binary icosahedral group it’s in

    • J. Ebert, The icosahedral group and the homotopy of the stable mapping class group

    diff, v26, current

    • CommentRowNumber41.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2021

    I have added some of the references on the group cohomology of the finite subgroups of SU(2)SU(2) given on p. 12 of Epa & Ganter 16

    Mainly that’s

    whose section 4 gives a fully detailed and completely explicit computation of the group cohomologies (including ring structure) of the three exceptional cases. That’s great to see.

    Then also

    but that just concerns the vanishing of the integral cohomology in degree 3.

    On top of that they point to Artin & Tate, but I haven’t found the relevant page and verse there yet.

    Last not least, they point to three paragraphs in Cartan & Eilenberg. My copy of that book loads slowly and is thus a pain to read. But from what I gleaned these three pointers are not so much a reference to the result, as an outline of how to go about proving it.

    diff, v32, current

    • CommentRowNumber42.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2021

    I have added (here) statement of the main result of Epa & Ganter 16, slighly recast, in this form:

    H 4(BSU(2),ℤ) ≃ ℤ Bi *↓ ↓ quotientcoprojection H 4(BG ADE,ℤ) ≃ ℤ/|G ADE|. \array{ H^4(B SU(2), \mathbb{Z}) &\simeq& \mathbb{Z} \\ {}^{\mathllap{B i^\ast}}\big\downarrow && \big\downarrow {}^{\mathrlap{ {quotient} \atop {coprojection} }} \\ H^4(B G_{ADE}, \mathbb{Z}) &\simeq& \mathbb{Z}/\left\vert G_{ADE}\right\vert \mathrlap{\,.} }

    diff, v33, current

    • CommentRowNumber43.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2021

    added pointer to:

    • Marjorie Senechal, Finding the Finite Groups of Symmetries of the Sphere, The American Mathematical Monthly 97 4 (1990) 329-335 (jstor:2324519)

    diff, v37, current

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