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8. Subfactors

    1.     The supertransitivity with respect to an object $X$ with $\dim(X)>2$ is the largest $N$ such that $\text{Hom}(1,X^{\otimes n})$ is Temperley-Lieb.

      Supertransitivity is the analog of transitivity of group actions.

      Problem 8.1.

      [Jones] Is there an upper bound on the supertransitivity of a subfactor planar algebra?
          Note that the group case was solved by the classification of finite simple groups.
        1. Remark. [org.aimpl.user:scott@tqft.net] Currently the extended Haagerup subfactor holds the record, with $n=7$. The Asaeda-Haagerup subfactor has $n=5$, and otherwise all known examples have $n\leq 4$.
            • Problem 8.2.

              [Snyder] Find a non-number theoretic argument to rule out the rest of the Haagerup family vine.

              For example, is there a diagram that evaluates in two different ways?
                • Problem 8.3.

                  [Morrison, Peters] Is there a polymer theory of principal graphs? What graphs can appear as subgraphs of principal graphs?
                      One would need a bound on the rank at each depth.

                  Penneys: Note that there are examples of forbidden subgraphs, e.g., part of the bad seed.

                      Cite this as: AimPL: Classifying fusion categories, available at http://aimpl.org/fusioncat.