An obstruction to subfactor principal graphs from the graph planar algebra embedding theorem

Abstract

We find a new obstruction to the principal graphs of subfactors. It shows that in a certain family of 3-supertransitive principal graphs, there must be a cycle by depth 6, with one exception, the principal graph of the Haagerup subfactor.

The classification of subfactors with index at most $5 \frac{1}{4}$

“…By Lemma 6.16, t ≤ 2. By Proposition 6.17 and the classi cation of subfactors to index 5, we must have that t = 0, but by [Mor14], we have t > 0, a contradiction.…”

“…However, we need a di erent way to deal with t = 0. The recent 3-supertransitive * 10 obstruction of [Mor14] does the trick.…”

“…Thus Lemma 6.20 actually eliminates all 2t-translated extensions of G with t ≥ 2! Again, the case t = 0 is ruled out by [Mor14], so the only remaining case is t = 1. Similar to the proof of Proposition 6.17, we need only check ω = 1, e ±2πi/3 by [Jon12, Theorem 5.1.11].…”

“…Two graphs (namely (f)(7) and (f)(8) in Theorem 3.1) require additional arguments in Section 6.3 using doubly one-by-one connection entries. We note that both these exceptional * 10 weeds require using Morrison's hexagon obstruction [Mor14] in an essential way. Of the remaining * 10 weeds, we get one candidate graph (from the rst graph in either (f)(3) or (f)(4) from Theorem 3.1) which is eliminated using the formal codegree obstruction (see Section 6.5).…”

Block maps and Fourier analysis

“…The region between the pair is colored by a third shading. Moreover, the principal graph and the graph planar algebra become refined [25,18]. From this point of view, we can represent the planar diagram D by the Bisch-Jones diagram shown in Fig 1 , when B(T, J) is a biprojection.…”