SpaceGroupIrep: A package for irreducible representations of space group

We have developed a Mathematica program package SpaceGroupIrep which is a database and tool set for irreducible representations (IRs) of space group in BC convention, i.e. the convention used in the famous book “The mathematical theory of symmetry in solids” by C.J. Bradley & A.P. Cracknell. Using this package, elements of any space group, little group, Herring little group, or central extension of little co-group can be easily obtained. This package can give not only little-group (LG) IRs for any k-point but also space-group (SG) IRs for any k-stars in intuitive table form, and both single-valued and double-valued IRs are supported. This package can calculate the decomposition of the direct product of SG IRs for any two k-stars. This package can 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. This package can also 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 very useful for both study and research, e.g. for analyzing band topology or determining selection rules.

本研究团队研发了一套名为SpaceGroupIrep的Mathematica程序包，该程序包集数据库与工具集于一体，旨在处理空间群在BC公约下的不可约表示（IRs）。换言之，该公约系著名书籍《固体对称性的数学理论》一书中所采用的公约。借助本程序包，用户可轻松获取任意空间群、小群、赫林小群或小共群中心扩展的元素。本程序包不仅能够提供任意k点的单值和双值小群（LG）IRs，还能以直观的表格形式呈现任意k星的空间群（SG）IRs。此外，本程序包还能计算任意两个k星空间群IRs直接乘积的分解。该程序包还能够确定在BC公约下能量带中布洛赫态的小群IRs，且无论输入原胞为何种类型，均能转换至BC公约下的单元，从而确保其适用性。此外，本程序包还能提供BCS（比利亚诺瓦晶体学服务器）与BC公约间k点和LG IR标签的对应关系。总而言之，SpaceGroupIrep程序包对于学术研究和研究工作极为有用，例如在分析带拓扑或确定选择规则方面。