MSGCorep: A package for corepresentations of magnetic space groups

Motivated by easy access to complete corepresentation (corep) data of all the 1651 magnetic space groups (MSGs) in three-dimensional space, we have developed a Mathematica package MSGCorep to provide an offline database of coreps and various functions to manipulate them, based on our previous package SpaceGroupIrep. One can use the package MSGCorep to obtain the elements of any MSG and magnetic little group, to calculate the multiplication of group elements, to obtain the small coreps at any k-point and full coreps of any magnetic k-star for any MSG and show them in a user-friendly table form, to calculate and show the decomposition of direct products of full coreps between any two specified magnetic k-stars, and to determine the small coreps of energy bands. Both single-valued and double-valued coreps are supported. In addition, the 122 magnetic point groups (MPGs) and their coreps are also supported by this package. To the best of our knowledge, MSGCorep is the first package that is able to calculate the direct product of full coreps for any MSG and able to determine small coreps of energy bands for general purpose. In a word, the MSGCorep package is an offline database and tool set for MSGs, MPGs, and their coreps, and it is very useful to study the symmetries in magnetic (type-I, -III, and -IV MSGs) and nonmagnetic (type-II MSGs) materials.为便捷获取三维空间中全部1651个磁空间群（magnetic space groups，MSGs）的完备共表示（corepresentation，简称corep）数据，我们基于此前开发的SpaceGroupIrep工具包，研发了Mathematica工具包MSGCorep，用于提供共表示的离线数据库以及各类共表示操作函数。用户可通过MSGCorep获取任意磁空间群与磁小群的群元、计算群元乘法；可获取任意磁空间群在任意k点处的小共表示，以及任意磁k星的完备共表示，并以用户友好的表格形式展示结果；可计算并展示任意两个指定磁k星之间的完备共表示直积的分解结果；还可求解能带的小共表示。该工具包同时支持单值与双值共表示。此外，它还涵盖了全部122个磁点群（magnetic point groups，MPGs）及其共表示的相关功能。据我们所知，MSGCorep是首个可针对任意磁空间群计算完备共表示直积、并可通用求解能带小共表示的工具包。总而言之，MSGCorep是面向磁空间群、磁点群及其共表示的离线数据库与工具集，可用于研究磁性（I型、III型及IV型磁空间群）与非磁性（II型磁空间群）材料中的对称性问题。