The construction of a $E_7$-like quantum subgroup of $SU(3)$

TL;DR

This paper constructs a specific quantum subgroup of SU(3) related to E7, embedding its planar algebra into a graph planar algebra, and identifies associated module categories and subfactor principal graphs.

Contribution

It introduces a new embedding of the planar algebra for a quantum group at a root of unity into a graph planar algebra, constructing a rank 11 module category over the quantum group.

Findings

Embedded the planar algebra into the graph planar algebra of  Francesco and Zuber's graph.

Constructed a rank 11 module category over  quantum group.

Determined principal graphs of certain subfactors.

Abstract

In this short note we construct an embedding of the planar algebra for $\overline{\operatorname{Rep}(U_q(sl_3))}$ at $q = e^{2\pi i \frac{1}{24}}$ into the graph planar algebra of di Francesco and Zuber's candidate graph $\mathcal{E}_4^{12}$. Via the graph planar algebra embedding theorem we thus construct a rank 11 module category over $\overline{\operatorname{Rep}(U_q(sl_3))}$ whose graph for action by the vector representation is $\mathcal{E}_4^{12}$. This fills a small gap in the literature on the construction of $\overline{\operatorname{Rep}(U_q(sl_3))}$ module categories. As a consequence of our construction, we obtain the principal graphs of subfactors constructed abstractly by Evans and Pugh.