GroupMath: A Mathematica package for group theory calculations

Abstract

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 summaryProgram Title: GroupMath: A Mathematica package for group theory calculationsCPC Library link to program files:https://doi.org/10.17632/hdkksr6v7t.1Developer's repository link:renatofonseca.net/groupmathLicensing provisions: GNU General Public License 3Programming language: MathematicaNature 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 Sm 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.