Finite flavour groups of fermions

TL;DR

This paper reviews the mathematical theory of finite groups and their application as flavour symmetries in particle physics, focusing on group structure, representations, and specific subgroups relevant to the field.

Contribution

It provides a comprehensive overview of finite groups, especially subgroups of SO(3) and SU(3), with emphasis on methods for exploring their properties in particle physics.

Findings

Detailed descriptions of finite subgroups of SO(3) and SU(3)

Analysis of conjugacy classes and character tables

Discussion on relationships between finite subgroups of U(3) and SU(3)

Abstract

We present an overview of the theory of finite groups, with regard to their application as flavour symmetries in particle physics. In a general part, we discuss useful theorems concerning group structure, conjugacy classes, representations and character tables. In a specialized part, we attempt to give a fairly comprehensive review of finite subgroups of SO(3) and SU(3), in which we apply and illustrate the general theory. Moreover, we also provide a concise description of the symmetric and alternating groups and comment on the relationship between finite subgroups of U(3) and finite subgroups of SU(3). Though in this review we give a detailed description of a wide range of finite groups, the main focus is on the methods which allow the exploration of their different aspects.