GroupMath is a Mathematica package which performs several calculations related to semi-simple Lie algebras and the permutation groups, both of which are important in various areas of physics. Having in mind the specific needs of theoretical particle physicists, the program computes several basis-independent quantities (such as roots, weights, decomposition of products of representations, branching rules) as well as basis-dependent ones (including explicit representation matrices and Clebsch-Gordon coefficients).

Program summary

Program Title: GroupMath: A Mathematica package for group theory calculations

CPC Library link to program files: https://doi.org/10.17632/hdkksr6v7t.1

Developer's repository link: renatofonseca.net/groupmath

Licensing provisions: GNU General Public License 3

Programming language: Mathematica

Nature of problem: Computations involving semi-simple Lie algebras are commonplace in several areas of research, such as particle physics. Indeed model builders trying to extend the Standard Model often need to consider new fields which transform as irreducible representations of these algebras. It is therefore convenient to have a computer program which can perform these calculations systematically. Such code might be used directly by the user, or be incorporated in a longer chain of programs. The permutation groups $S_m$ are also important, both by themselves and also as a tool to better describe the properties of the representations of semi-simple Lie algebras.

Solution method: GroupMath performs several computations related to semi-simple Lie algebras which are important in high energy physics: Cartan matrices, roots, weights, Weyl reflections, decomposition of products of representations, subgroups, branching rules, and others. Algorithms available in the literature are used to efficiently perform some of these calculations. The program also computes basis-dependent quantities such as representation matrices and Clebsch-Gordon coefficients. Several elements of the theory of finite permutation groups have also been implemented, including Young diagrams and tableaux, the decomposition of products of representations, branching rules, and explicit representation matrices.

Additional comments including restrictions and unusual features: Most of GroupMath's functions can handle arbitrary representations of any semi-simple Lie algebra and any permutation group. However, memory and time constraints are important when considering large and/or numerous representations. This is particularly true for the functions which perform basis-dependent computations.

中文翻译：

---

GroupMath：用于群论计算的 Mathematica 包

GroupMath 是一个 Mathematica 包，它执行一些与半简单李代数和置换群相关的计算，这两者在物理学的各个领域都很重要。考虑到理论粒子物理学家的特定需求，该程序计算了几个与基础无关的量（例如根、权重、表示积的分解、分支规则）以及与基础相关的量（包括显式表示矩阵和 Clebsch-戈登系数）。

程序概要

程序名称： GroupMath：用于群论计算的 Mathematica 包

CPC 库程序文件链接： https : //doi.org/10.17632/hdkksr6v7t.1

开发人员的存储库链接： renatofonseca.net/groupmath

许可条款：GNU 通用公共许可证 3

编程语言： Mathematica

问题的性质：涉及半简单李代数的计算在多个研究领域都很常见，例如粒子物理学。事实上，试图扩展标准模型的模型构建者通常需要考虑新的领域，这些领域将转化为这些代数的不可约表示。因此，拥有一个可以系统地执行这些计算的计算机程序是很方便的。此类代码可能由用户直接使用，或者合并到更长的程序链中。置换群${秒}_{米}$也很重要，无论是自身还是作为更好地描述半简单李代数表示的性质的工具。

求解方法： GroupMath 执行一些与半简单李代数相关的计算，这些计算在高能物理中很重要：嘉当矩阵、根、权重、外尔反射、表示的积分解、子群、分支规则等。文献中可用的算法用于有效地执行其中一些计算。该程序还计算与基础相关的量，例如表示矩阵和 Clebsch-Gordon 系数。还实施了有限置换群理论的几个要素，包括杨氏图和表格、表示积的分解、分支规则和显式表示矩阵。

附加说明，包括限制和不寻常的特征：大多数 GroupMath 函数可以处理任何半简单李代数和任何置换群的任意表示。但是，在考虑大型和/或众多表示时，内存和时间限制很重要。对于执行依赖于基础的计算的函数来说尤其如此。