The $E_7$ Commuting Squares Produce $D_{10}$ as Principal Graph

Abstract

We prove that the (two) connections, or commuting squares, on the Coxeter–Dynkin diagram $E_7$ produce a subfactor with principal graph $D_{10}$. This was conjectured by J.-B. Zuber in connection with modular invariants in conformal field theory, and solve the last case of computing the flat parts of the connections on Coxeter–Dynkin diagrams with index less than 4.