Composed inclusions of $A_3$ and $A_4$ subfactors

TL;DR

This paper classifies all possible standard invariants arising from the composition of an A_3 and an A_4 subfactor, providing a complete answer to a longstanding question in subfactor theory.

Contribution

It identifies exactly four standard invariants for the composed inclusion of A_3 and A_4 subfactors, resolving a question posed in 1994.

Findings

Four standard invariants from A_3 and A_4 composition

Complete classification of invariants for composed A_4 subfactors

Answer to a 1994 open problem

Abstract

In this article, we classify all standard invariants that can arise from a composed inclusion of an $A_3$ with an $A_4$ subfactor. More precisely, if $\mathcal{N}\subset \mathcal{P}$ is the $A_3$ subfactor and $\mathcal{P}\subset\mathcal{M}$ is the $A_4$ subfactor, then only four standard invariants can arise from the composed inclusion $\mathcal{N}\subset\mathcal{M}$. This answers a question posed by Bisch and Haagerup in 1994. The techniques of this paper also show that there are exactly four standard invariants for the composed inclusion of two $A_4$ subfactors.