A new set of generators and a physical interpretation for the SU(3) finite subgroup D(9,1,1;2,1,1)

TL;DR

This paper introduces new generators and a physical interpretation for a specific finite subgroup of SU(3), enhancing understanding of its structure and connection to braid group representations and Chern-Simons theory.

Contribution

It provides a minimal set of generators for the SU(3) subgroup D(9,1,1;2,1,1) and reveals its isomorphism to a semidirect product, linking it to braid group representations.

Findings

New generators for the subgroup D(9,1,1;2,1,1) of SU(3)

Identification of the subgroup's structure as Z_9 × Z_3 ⋊ S_3

Connection of generators to SU(2) Chern-Simons theory at level 4

Abstract

After 100 years of effort, the classification of all the finite subgroups of SU(3) is yet incomplete. The most recently updated list can be found in P.O. Ludl, J. Phys. A: Math. Theor. 44 255204 (2011), where the structure of the series (C) and (D) of SU(3)-subgroups is studied. We provide a minimal set of generators for one of these groups which has order 162. These generators appear up to phase as the image of an irreducible unitary braid group representation issued from the Jones-Kauffman version of SU(2) Chern-Simons theory at level 4. In light of these new generators, we study the structure of the group in detail and recover the fact that it is isomorphic to the semidirect product Z_9 \times Z_3 \rtimes S_3 with respect to conjugation.