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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 11th 2018

    Will start something here.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 11th 2018

    Some content

    v1, current

    • 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.

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeFeb 26th 2019

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

    1 1 normal subgrouplattice of translations N E translationgroup crystallographicgroup S Iso(E) Euclideanisometry group pointgroup G O(E) orthogonalgroup 1 1 \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 NN is what makes the action of the point group GG descend to the torus E/NE/N.

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

    diff, v3, current

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

    started to add various references.

    Added a long quote from

    • E. V. Chuprunov, T. S. Kuntsevich, nn-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.

    diff, v4, current

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

    added pointer to

    • 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 2 4\mathbb{C}^2 \simeq_{\mathbb{R}} \mathbb{R}^4, as point groups of crystallographic groups in 4d.

    diff, v5, current

    • 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 nD_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 NGN \rtimes G in 4d, with N= 4N = \mathbb{Z}^4 the canonical translational subgroup of 4\mathbb{R}^4 and the point group GG acting on 4 \mathbb{R}^4\simeq_{\mathbb{R}} \mathbb{H} via their inclusion as subgroups of Sp(1)Sp(1):

    NG=...N \rtimes G = ...

    1. 4 2\mathbb{Z}^4 \rtimes \mathbb{Z}_2

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

    3. 4Q 8\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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