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What kind of symmetry groups, analogue of 3D crystallographic space groups, can describe the spatial symmetries of 3D systems which have discrete periodicity in only one direction, but are homogeneous in the 2D transverse plane?

What is the classification and the representation theory of such kind of symmetry groups?

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If the system is only discretely periodic in one dimension, the only point groups are the trivial group $C_1$ and the group $Z_2$ generated by reflections about a point.

These, combined with the discrete translational symmetry of periodicity, give the 1D line group.

See https://en.wikipedia.org/wiki/One-dimensional_symmetry_group for more details.

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  • $\begingroup$ Despite the 1D discrete periodicity, the compatible rotation symmetries of such 3D systems are far more than the reflection in 1D point group. And the system also can have nonsymmorphic symmetries, i.e. glide reflections and screw rotations. Actually, 3D systems with 1D periodicity can be described by 3D line groups (en.wikipedia.org/wiki/Line_group), but line groups do not include the continuous translation symmetry in the transverse plane. $\endgroup$ – zrysky Jun 13 '18 at 7:26
  • $\begingroup$ My apologies; I misread "space group" as "point group". The constraint of homogeneity in the transverse direction restricts to the 1D line groups as far as I can tell, though? I have edited the answer to reflect this. $\endgroup$ – DavidH Jun 13 '18 at 8:49
  • $\begingroup$ It's not that simple. The homogeneity in the transverse plane does not restrict the symmetry to 1D line group, the system can support rotation symmetries around the periodic axis, and even nonsymmorphic symmetries. An example is the system of layered anisotropic dielectric slabs, the orientation of the optical axes of the dielectric slabs constrains the rotation symmetry around the axis. $\endgroup$ – zrysky Jun 14 '18 at 6:25

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