On Haagerup’s List of Potential Principal Graphs of Subfactors

Abstract

Abstract. We show that any graph, in the sequence given by Haagerup in 1991 as that of candidates of principal graphs of subfactors, is not realized as a principal graph except for the smallest two. This settles the remaining case of a previous work of the first author.

Subfactors of Index less than 5, Part 1: The Principal Graph Odometer

“…In Haagerup's paper he claims that that in the Asaeda-Haagerup family only supertransitivity 5 is possible. Since Haagerup's paper the hexagon family has been entirely ruled out [Bis98] and the Haagerup family at supertransitivities 11 and above have been ruled out [Asa07,AY09]. A uniform argument for all three cases (and indeed excluding all but finitely many examples coming from any vine) can be given using [CMS11].…”

“…Number theory techniques of Calegari-Morrison-Snyder [CMS11] (generalizing earlier work of Asaeda-Yasuda [AY09]) give an effective bound on how large a translate of a fixed graph can be a principal graph. Thus any classification along the lines of Haagerup's can now be reduced to a finite list by applying this result.…”

“…Haagerup eliminated all but one graph in the second family using an unpublished connection-based approach, Bisch eliminated the third family completely using fusion algebra techniques [Bis98], and Asaeda-Yasuda [AY09] eliminated all but two graphs from the first family using arithmetic techniques.…”

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

“…Moreover, with sufficiently powerful combinatorial (and number theoretic, cf. [1,3,5,19]) constraints in hand, it has proved possible to enumerate all possible principal graphs for subfactors with small index. This approach was pioneered by Haagerup [7], and more recently continued, resulting in a classification of subfactors up to index 5 [8,11,15,17,21].…”

Subfactors of Index Less Than 5, Part 4: Vines

“…They prove that given a vine (Γ, v), one can compute an N (Γ) ∈ N such that Γ n is not a principal graph for all n > N (Γ), where Γ n is the translation of Γ at v with n vertices. A similar approach was used in [AY09] to eliminate translates of the Haagerup family (Vines 7 and 9) as possible principal graphs.…”