SpaceGroupIrep: A package for irreducible representations of space group

TL;DR

SpaceGroupIrep is a Mathematica package that provides comprehensive tools and data for analyzing irreducible representations of space groups, aiding research in band topology and symmetry analysis.

Contribution

It introduces a new software package that simplifies obtaining and analyzing IRs of space groups in BC convention, supporting various calculations and conversions.

Findings

Supports IRs for any k-point and k-star

Calculates decomposition of IR products

Converts between BCS and BC conventions

Abstract

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.