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    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 15th 2018

    I added a loose description of the dihedron, and commented that the 2-gon as a face should be possible (so as to have the A 1A_1 case included, thinking of the ADE classification)

    diff, v2, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2018

    Thanks. I made Platonic solid a hyperlink.

    (Let’s remember to hyperlink at least the key technical terms in an entry. That’s what make a wiki be more useful than a book.)

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 16th 2018

    Why is the index 4n4 n for the dihedral group D 4nD_{4n}? I have a vague recollection of a difference in terminology about say D 5D_5 or D 10D_10 for symmetries of the pentagon. Ah yes, wikipedia mentions this. But this concerns nn or 2n2n, not 4n4n.

    Another point, we claim that the ADE classification concerns Platonic solids, and yet don’t associate anything with the AA series in the table. Is there a way of associating degenerate solids to both AA and DD? Perhaps this page helps.

    I have a feeling there’s something else wrong with that table. Wikipedia speaks of a ’binary cyclic group’, which is what we should have presumably as a subgroup of SU(2)SU(2).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2018

    I have a feeling there’s something else wrong with that table. Wikipedia speaks of a ’binary cyclic group’, which is what we should have presumably as a subgroup of SU(2)SU(2).

    Thanks for catching that. I created binary cyclic group and fixed the ADE – table.

    But why “something else”? What else is wrong?

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 16th 2018

    The index in D 4nD_{4n} issue.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2018
    • (edited Apr 16th 2018)

    I see now. Thanks.

    Let me see. here is a corresponding table in Durfee 79

    It seems to say that

    Dynkin fin group symbol order
    D kD_k binary dihedral D kD_k 4(k-2)

    for k4k \geq 4.

    I would like us to start counting at 0. That should give

    Dynkin fin group symbol order
    D n+4D_{n+4} binary dihedral D n+4D_{n+4} 2(2n+4)

    for nn \in \mathbb{N}

    This seems to fit with neither of the two conventions that Wikipedia offers, even if one accounts for the binary version.

    But we get from it that the non-binary dihedral group corresponding to the Dynkin diagram D n+4D_{n+4} has order 2(n+2)2(n+2). If we follow Wikipedia, then this should be called either D n+2D_{n+2} or D 2n+4D_{2n+4}.

    What a mess!

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 16th 2018

    Gah, is it too much to ask for Wikipedia to give the collection of unit quaternions corresponding to the binary cyclic group? Cf https://groupprops.subwiki.org/wiki/Dicyclic_group (which I will add later if no one beats me to it).

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2018

    I have a feeling there’s something else wrong with that table. Wikipedia speaks of a ’binary cyclic group’, which is what we should have presumably as a subgroup of SU(2)SU(2).

    Thanks for catching that. I created binary cyclic group and fixed the ADE – table.

    Sorry, that was wrong. I changed it back. The non-binary 2n+1\mathbb{Z}_{2n+1} are still finite subgroups of SU(2)SU(2), of course: The generator is

    (e 2πi/(2n+1) 0 0 e 2πi/(2n+1)) \left( \array{ e^{2\pi i / (2n + 1)} & 0 \\ 0 & e^{-2\pi i / (2n + 1)} } \right)
    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 16th 2018

    So the odd cyclics are subgroups of SU(2)SU(2) but are not binary polyhedral groups.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2018

    Yes.

    In Keenan 03, theorem 4 it is phrased this way:

    Every finite subgroup of SU(2)SU(2) is a cyclic, binary dihedral or binary polyhedral group.

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 16th 2018
    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeApr 17th 2018

    That pointer should go to binary dihedral group. I have added it there.

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 17th 2018

    Thanks. I was looking up stuff on my phone while walking to the bus, not quite up to editing a lab page.

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