On the fusion algebras of bimodules arising from Goodman-de la Harpe-Jones subfactors

TL;DR

This paper provides a combinatorial method to determine the fusion rules and dual principal graphs of bimodules from Goodman-de la Harpe-Jones subfactors, including the exceptional E8 case, and explores subequivalence among A-D-E paragroups.

Contribution

It introduces a purely combinatorial approach to compute fusion rules and dual principal graphs for bimodules from Goodman-de la Harpe-Jones subfactors, extending to the E8 case.

Findings

Derived dual principal graphs and fusion rules combinatorially.

Explicitly obtained the E8 subfactor's fusion data.

Identified subequivalence relations among A-D-E paragroups.

Abstract

By using Ocneanu's result on the classification of all irreducible connections on the Dynkin diagrams, we show that the dual principal graphs as well as the fusion rules of bimodules arising from any Goodman-de la Harpe-Jones subfactors are obtained by a purely combinatorial method. In particular we obtain the dual principal graph and the fusion rule of bimodules arising from the Goodman-de la Harpe-Jones subfactor corresponding to the Dynkin diagram $E_8$. As an application, we also show some subequivalence among $A$-$D$-$E$ paragroups.