The Freedman group: a physical interpretation for the SU(3)-subgroup D(18,1,1;2,1,1) of order 648

Abstract

We study a subgroup $Fr(162\times 4)$ of SU(3) of order 648 which is an extension of $D(9,1,1;2,1,1)$ and whose generators arise from anyonic systems. We show that this group is isomorphic to a semi-direct product $(\mathbb{Z}/18\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})\rtimes S_3$ with respect to conjugation and we give a presentation of the group. We show that the group $D(18,1,1;2,1,1)$ from the series $(D)$ in the existing classification for finite SU(3)-subgroups is also isomorphic to a semi-direct product $(\mathbb{Z}/18\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})\rtimes S_3$, also with respect to conjugation. We show that the two groups $Fr(162\times 4)$ and $D(18,1,1;2,1,1)$ are isomorphic and we provide an isomorphism between both groups. We prove that $Fr(162\times 4)$ is not isomorphic to the exceptional SU(3) subgroup $\Sigma(216\times 3)$ of the same order 648. We further prove that the only SU(3) finite subgroups from the 1916 classification by Blichfeldt or its extended version which $Fr(162\times 4)$ may be isomorphic to belong to the $(D)$-series. Finally, we show that $Fr(162\times 4)$ and $D(18,1,1;2,1,1)$ are both conjugate under an orthogonal matrix which we provide.