Notes about this project and about layer and line groups

Symmetry is a profound property of a physical system, determining most of its physical characteristics. Low-dimensional (D) crystals have attracted great attention of physicists and material scientists during the past three decades. From the point of view of symmetry, this raised the interest for line and layer (diperiodic) groups, related to quasi-1D and quasi-2D regular compounds. While space groups, symmetries of 3D crystals, are well elaborated, with a number of crystallographic references, books and sites, the low-D structures are not as thoroughly covered. In fact, there is a monograph on crystallographic sub-periodic groups, without k-space data (thus without irreducible representations), as well as the book of Litvin on magnetic crystallographic groups. For line groups this means that only rod groups ( eighty of continuously many line groups – mostly non-crystallographic) are elaborated. This is partly compensated by a number of papers and the reference book . As for the layer groups, many data can be obtained from their space supergroups; still this is a cumbersome task, and insufficient for some properties. In addition, there is a Evarestov and Smirnov book and the paper of ours on the layer groups irreducible representations.

The information in the electronic form is advantageous in several views: it is not bounded in details or volume, it is much more transparent and easier to search, and immediately available to community. This is many years ago noticed by the researchers running
Bilbao crystallographic server
which is nowadays one of the most popular tools for condensed matter and material science researchers, especially those understanding and using group theory. Although the idea proved to be useful, and although the Bilbao site is well organized for space groups, the line groups are completely missing (besides some data for rod groups), and as for the layer groups, the information is incomplete. Therefore, we aim at filling this gap, giving exhaustive database both for line and layer groups.

In the background of this site is the computer program POLSym (originally aimed for polymer symmetry), which has been developing in NanoLab for 25 years. It is symbolic algebra code (Wolfram Mathematica), which generated literally all the data presented at the site. Its main advantage is full implementation of the
modified group projector theory, which uses factorization of the point, line and layer groups into weak-direct product of cyclic subgroups, to enable manipulation with the group generators only (avoiding summation over group and similar complications in numerical approach). Accordingly, symbolic algebra offers the quite general form of irreducible (and allowed) representations (with k-dependent matrices) and derived quantities, as well as reduction of induced representations, or construction of elementary band representations (including topologically diferent graphs). Also, all manipulation with linear-antilinear representations of magnetic (gray or black-and-white groups) or double groups are straightforwardly enabled.

It should be stressed out that the site will be gradually filled with new, already POLSym prepared data. We started with elementary data related to 320 (single and double, ordinary and gray) layer groups. Subsequently, black and white groups are planned to be included, firstly. Then, the other data (e.g. topologically classified elementary band representations) are to follow. The timing depends on our web-site designing skills (we expect grant to employ web-site designer), publication duties and other research projects of Nanolab. For the mentioned reasons the web site has very simple structure, with static tables (and minimal entry description), but we hope to give more contextual explanations, direct online calculation and other web facilities. Suggestions and comments are welcome at

Although NanoLab is primary interested in line groups and physical properties of the quasi-1D regular structures (nanotubes, nanowires, polymers), the enormous development of research of layers, and in particular symmetry based topological analysis of their band structures motivated us to start the low-D symmetry enterprise by including also the layer groups. Actually, the site devoted to the line groups is already conceptually designed, and in near future, the part of site devoted to the line groups will be made visible.