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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 $A_1$ case included, thinking of the ADE classification)

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 16th 2018

(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 $4 n$ for the dihedral group $D_{4n}$? I have a vague recollection of a difference in terminology about say $D_5$ or $D_10$ for symmetries of the pentagon. Ah yes, wikipedia mentions this. But this concerns $n$ or $2n$, not $4n$.

Another point, we claim that the ADE classification concerns Platonic solids, and yet don’t associate anything with the $A$ series in the table. Is there a way of associating degenerate solids to both $A$ and $D$? 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 $SU(2)$.

• 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 $SU(2)$.

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 $D_{4n}$ 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
$D_k$ binary dihedral $D_k$ 4(k-2)

for $k \geq 4$.

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

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

for $n \in \mathbb{N}$

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 $D_{n+4}$ has order $2(n+2)$. If we follow Wikipedia, then this should be called either $D_{n+2}$ or $D_{2n+4}$.

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 $SU(2)$.

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 $\mathbb{Z}_{2n+1}$ are still finite subgroups of $SU(2)$, of course: The generator is

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

So the odd cyclics are subgroups of $SU(2)$ 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 $SU(2)$ 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.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeAug 29th 2019

In the sentence

A dihedron is a degenerate Platonic solid with only two (identical) faces, which may be any polygon (including possibly the degenerate

I have replaced “polygon” by “regular polygon”