Principal graph stability and the jellyfish algorithm

TL;DR

This paper establishes a connection between principal graph stability in subfactor planar algebras and the existence of jellyfish generators, characterizing when the principal graph is a spoke graph.

Contribution

It proves that stable principal graphs with certain conditions must end in A_{finite} tails and characterizes jellyfish generators via spoke graphs.

Findings

Stable principal graphs with two stable depths end in A_{finite} tails.

Jellyfish generators exist at depth n if and only if the principal graph is a spoke graph.

The results extend Popa's theorem on principal graph stability.

Abstract

We show that if the principal graph of a subfactor planar algebra of modulus \delta>2 is stable for two depths, then it must end in A_{finite} tails. This result is analogous to Popa's theorem on principal graph stability. We use these theorems to show that an (n-1) supertransitive subfactor planar algebra has jellyfish generators at depth n if and only if its principal graph is a spoke graph.