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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeDec 11th 2018

Will start something here.

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeDec 11th 2018

Some content

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeDec 11th 2018

Thanks!. I slightly edited the first sentence for links:

We do have Euclidean group, albeit just a stub.

And, due to the dominance of logicians around here, for better or worse our lattice points to the order-theoretic concept, while for the “common” meaning we need to type lattice in a vector space.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeFeb 26th 2019

expanded the Idea/Definition-section, in particular added this diagram:

$\array{ & 1 && 1 \\ & \downarrow && \downarrow \\ {\text{normal subgroup} \atop \text{lattice of translations}} & N &\subset& E & {\text{translation} \atop \text{group}} \\ & \big\downarrow && \big\downarrow \\ {\text{crystallographic} \atop \text{group}} & S &\subset& Iso(E) & {\text{Euclidean} \atop \text{isometry group}} \\ & \big\downarrow && \big\downarrow \\ {\text{point} \atop \text{group}} & G &\subset& O(E) & {\text{orthogonal} \atop \text{group}} \\ & \downarrow && \downarrow \\ & 1 && 1 }$

Then I added the remark (here) that the normality of the subgroup $N$ is what makes the action of the point group $G$ descend to the torus $E/N$.

Is there a good name for these tori $E/N$ equpped with actions of Euclidean point groups $G$? They want to be called “representation tori” to go along with representation spheres, but I guess nobody says that.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeFeb 26th 2019
• (edited Feb 26th 2019)

• E. V. Chuprunov, T. S. Kuntsevich, $n$-Dimensional space groups and regular point systems, Comput. Math. Applic. Vol. 16, No. 5-8, pp. 537-543, 1988 (doi:10.1016/0898-1221(88)90243-X)

on the history and current scope of the classification.

Not done yet, but need to interrupt now.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeAug 31st 2019
• (edited Aug 31st 2019)

• D. Weigel, T. Phan and R. Veysseyre, Crystallography, geometry and physics in higher dimensions. III. Geometrical symbols for the 227 crystallographic point groups in four-dimensional space, Acta Cryst. (1987). A43, 294-304 (doi:10.1107/S0108767387099367)

am trying to find out which of the finite subgroups of SU(2) in the D- and E-series arise, via their canonical action on $\mathbb{C}^2 \simeq_{\mathbb{R}} \mathbb{R}^4$, as point groups of crystallographic groups in 4d.

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeAug 31st 2019

Is it just a matter of going through a list? The list in ’Summary’ tells you which are non-crystallographic.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeAug 31st 2019

Yes, it should be. But I have trouble parsing the entries.

Does $D_n$ denote the dihedral group? Does the binary dihedral group appear anywhere?

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeAug 31st 2019

So I suppose I can easily check by direct inspection that the following are crystallographics groups $N \rtimes G$ in 4d, with $N = \mathbb{Z}^4$ the canonical translational subgroup of $\mathbb{R}^4$ and the point group $G$ acting on $\mathbb{R}^4\simeq_{\mathbb{R}} \mathbb{H}$ via their inclusion as subgroups of $Sp(1)$:

$N \rtimes G = ...$

1. $\mathbb{Z}^4 \rtimes \mathbb{Z}_2$

2. $\mathbb{Z}^4 \rtimes \mathbb{Z}_4$

3. $\mathbb{Z}^4 \rtimes Q_8$

So these must appear somewhere in these lists. But where? I must be missing something.

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeSep 1st 2019

The term ’binary dihedral group’ shows up on the 3d page, but not given in Coxeter notation. Coxeter notation seems confusing.

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