Principal series of finite subgroups of SU(3)

TL;DR

This paper provides a comprehensive analysis of the exceptional finite subgroups of SU(3), including their structure, representations, and tensor products, using principal series to simplify computations and understand their relationships.

Contribution

It offers a detailed description of the exceptional groups Sigma(36x3), Sigma(72x3), and Sigma(216x3), introducing the use of principal series for easier analysis and understanding of these groups.

Findings

Derived conjugacy classes, normal subgroups, and character tables.

Showed principal series simplifies computations and reveals group relationships.

Studied dihedral-like groups as a testing ground for principal series methods.

Abstract

We attempt to give a complete description of the "exceptional" finite subgroups Sigma(36x3), Sigma(72x3) and Sigma(216x3) of SU(3), with the aim to make them amenable to model building for fermion masses and mixing. The information on these groups which we derive contains conjugacy classes, proper normal subgroups, irreducible representations, character tables and tensor products of their three-dimensional irreducible representations. We show that, for these three exceptional groups, usage of their principal series, i.e. ascending chains of normal subgroups, greatly facilitates the computations and illuminates the relationship between the groups. As a preparation and testing ground for the usage of principal series, we study first the dihedral-like groups Delta(27) and Delta(54) because both are members of the principal series of the three groups discussed in the paper.