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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.
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.
started to add various references.
Added a long quote from
on the history and current scope of the classification.
Not done yet, but need to interrupt now.
added pointer to
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.
Is it just a matter of going through a list? The list in ’Summary’ tells you which are non-crystallographic.
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?
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 $Sp(1)$:
$N \rtimes G = ...$
$\mathbb{Z}^4 \rtimes \mathbb{Z}_2$
$\mathbb{Z}^4 \rtimes \mathbb{Z}_4$
$\mathbb{Z}^4 \rtimes Q_8$
So these must appear somewhere in these lists. But where? I must be missing something.
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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