A mathematica package for irreducible representations of space group in BC convention. Here, "BC convention" means the convention used in the the famous book “The mathematical theory of symmetry in solids” by C. J. Bradley & A. P. Cracknell (called the BC book hereafter).

• Users can join WeChat group for communications: the QR code. If you cannot join the WeChat group, please contact me through gbliu(AT)bit(dot)edu(dot)cn.
• 点击上面链接加入SpaceGroupIrep使用交流微信群，如无法加入可邮件联系。

• Comput. Phys. Commun. 265 , 107993 (2021) (arXiv:2012.08871)

If you use this package in your research, please cite the above paper.

• Elements of any space group, little group, Herring little group, or central extension of little co-group can be easily obtained.
• Identify k-points. Support the cases in which one k-point may have two different names for different shapes of Brillouin zones due to different ratios of lattice constants.
• Show the little-group (LG) irreducible representations (IRs) for any k-point in intuitive table form,.
• Show the space-group (SG) IRs for any k-stars in intuitive table form.
• Both single-valued and double-valued IRs are supported.
• Calculate the decomposition of the direct product of SG IRs for any two k-stars.
• Determine the LG IRs of Bloch states in energy bands in BC convention. And this works for any input primitive cell thanks to its ability to convert any input cell to a cell in BC convention.
• Provide the correspondence of k-points and LG IR labels between BCS (Bilbao Crystallographic Server) and BC conventions.

In a word, the package SpaceGroupIrep is a database and tool set for IRs of space group in BC convention, and it is very useful for both study and research.

• SpaceGroupIrep.wl The main file containing most functions and data.
• AbstractGroupData.wl The abstract group data in BC-Tab. 5.1 (meaning the Tab. 5.1 in the BC book).
• LittleGroupIrepData.wl This file contains the data of LG IRs in BC-Tabs. 5.7, 5.11, 6.13, and 6.15.
• allBCSkLGdat.mx This file contains the BCS data of LG IRs collected from the output of irvsp.
• examples/test.nb A simple test file or demo file.
• examples/document and examples.nb A more detailed example file including some explanations.

Place the directory SpaceGroupIrep containing at least the four files, i.e. SpaceGroupIrep.wl, AbstractGroupData.wl, LittleGroupIrepData.wl, and allBCSkLGdat.mx, under any of the following paths:

• $InstallationDirectory/AddOns/Packages/
• $InstallationDirectory/AddOns/Applications/
• $BaseDirectory/Applications/
• $UserBaseDirectory/Applications/

where $InstallationDirectory is the installation directory of Mathematica (version ≥ 11.2), and $BaseDirectory and $UserBaseDirectory are the directories containing respectively systemwide and user-specific files loaded by Mathematica. The concrete values of $InstallationDirectory, $BaseDirectory, and $UserBaseDirectory can be obtained by running them in Mathematica because they are all built-in symbols.

Then one can use the package after running <<"SpaceGroupIrep`".

In the supplementary material of the paper Comput. Phys. Commun. 265 , 107993 (2021) we list the typos found in the BC book. However, I found new typos in the BC book after the publication of the paper. They are listed here:

• In BC-Tab 5.1, for the abstract group $G_{96}^4$ (p.271), the generator $S$ for reps R7, R8, and R9 is wrong because it does not satisfy $S^2=P^3$. There should be $S=\kappa$ for them all. This affects the single-valued LG IR of $H$ point of the No. 230 space group.
• In BC-Tab 5.1, for the abstract group $G_{192}^2$ (p.281), the generator $S$ of R18 does not satisfy $S^2=E$. There should be $S={{0,\varepsilon},{\varepsilon,0}}$. This affects the double-valued LG IR of $R$ point of the Nos. 222 and 223 space groups and $H$ point of the No. 230 space group.