Consider the following process:
- Take a maximal finite depth-index irreducible subfactor planar algebra $P^{(1)} = P(A^{(0)} \subset A^{(1)})$.
- Choose a composition with itself such that there is no extra biprojection and the depth is minimal.
- Iterate the process for getting a chain without extra intermediate and at each step the composition is chosen with minimal depth: $A^{(0)} \subset A^{(1)} \subset \dots A^{(n)}$ and let $P^{(n)} = P(A^{(0)} \subset A^{(n)})$
- For all $n$, choose a minimal projection $u_n \in P^{(n)}_{2,+}$ generating the identity biprojection.
- Let the planar algebra $Q=\bigotimes_n P^{(n)}$ (up to some technical justifications)
- Take $v \in Q_{2,+}$ the projection $u_1 \otimes u_2 \otimes \dots \otimes u_n \otimes \dots $
- Let $b$ be the biprojection generated by $v$ and $b_m$ the biprojection generated by $v^{*m}$
- Get the subfactor planar algebra corresponding to $(b_m \le b)$.
Conclusion: from one maximal subfactor $(A^{(0)} \subset A^{(1)})$, we get a series of new subfactors.
Example: by starting with the $TLJ$ subfactor planar algebra $A_3$ (i.e. the order $2$ cyclic group subfactor), this process produces every order $m$ cyclic group subfactor planar algebra, because: $$m \mathbb{Z} \subset \mathbb{Z} \subset G=\prod_{n=1}^{\infty} \mathbb{Z}/2^{n}\mathbb{Z}$$ In fact the element $g=(1,1, \dots) \in G$ generates the subgroup $\langle g \rangle =\mathbb{Z} \supset m\mathbb{Z} = \langle g^{m} \rangle $
Question: Is this process working in general (up to some technical justifications)?
Else, what are the main obstructions?
Question: What's happening by starting with $A_4$? Is this producing new subfactors?
The compositions of two $A_4$ are classified in this paper.
Next, what's happening by starting with $A_n$ for a given $n \ge 5$?
Remark: For $A_5$, ie $(D_1 \subset D_3)$, we get infinite index subfactors which are not new (see here).
In this case, a generalization could be useful.
Optional part: Generalization
Let $(N \subset M)$ be a finite depth-index irreducible subfactor, and let $P = P(N \subset M)$
Definition: A minimal projection $u \in P_{2,+}$ is order $r$ if $e_N \preceq u^{*r}$ and $e_N \not\preceq u^{*s}$ for $0<s<r$.
The previous process admits the following generalization:
Let $P^{(1)}, P^{(2)}, P^{(3)}, \dots$ be a sequence of finite depth-index irreducible subfactor planar algebras such that for all $n$ there is a minimal projection $u_n \in P^{(n)}_{2,+}$ such that the order of $u_n$ is strictly increasing. Take $Q=\bigotimes_n P^{(n)}$, $v=\bigotimes_n u_n \in Q_{2,+}$, $b=\langle v \rangle$ and $b_m=\langle v^{*m} \rangle$, then for every $m$ we obtain the subfactor related to $(b_m \le b)$.
