Fusion rules on a parametrized series of graphs

TL;DR

This paper investigates a series of graph pairs related to subfactors, establishing the existence of a unique compatible fusion system for all non-negative integers, despite only some pairs corresponding to actual subfactors.

Contribution

It proves the existence and uniqueness of a fusion system for a parametrized series of graph pairs, extending understanding beyond known subfactor cases.

Findings

Unique fusion system exists for all graph pairs in the series.

Only the first two pairs correspond to actual subfactors.

Fusion system compatibility is established for all non-negative integers.

Abstract

A series of pairs of graphs (Gamma_k, Gamma'_k), k = 0,1,2,... has been considered as candidates for dual pairs of principal graphs of subfactors of small Jones index above 4 and it has recently been proved that the pair (Gamma_k, Gamma'_k) comes from a subfactor if and only if k = 0 or k =1. We show that nevertheless there exists a unique fusion system compatible with this pair of graphs for all non-negative integers k.