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

Abstract

In this short note we construct an embedding of the planar algebra for $\overline{\mathrm{Rep}(U_q(s{l}_3))}$ 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 $\overline{\mathrm{Rep}(U_q(s{l}_3))}$ 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 $\overline{\mathrm{Rep}(U_q(s{l}_3))}$ module categories. As a consequence of our construction, we obtain the principal graphs of subfactors constructed abstractly by Evans and Pugh.

9 pages, Minor changes to address the referee's report, added Mathematica file with solutions & verification of main theorem