Let $N \subset M$ be an irreducible finite depth and finite index subfactor.
$M$ is a completely reducible (algebraic) $N$-$N$ bimodule, it decomposes into irreducibles as follows : $$M=\bigoplus_{i \in I} V_i \otimes M^{(i)}$$ with $(M^{(i)})_{i \in I}$ the irreducible bimodules, $M^{(0)}=N$, and $(V_i)_{i \in I}$ the multiplicity spaces.
Definition : such a subfactor has property $C$ if : $\exists i \in I$, $ \exists v \in V_i$ such that $\langle N, v \otimes M^{(i)} \rangle=M$
Question : how translate this property into the planar algebra framework ?
Remarks : $M^{(i)}$ is an irreducible $N$-$N$ bimodule, so $\forall x \in M^{(i)}$, $\langle N, v \otimes x \rangle=\langle N, v \otimes M^{(i)} \rangle$.
I suspect that property $C$ characterizes what I call the cyclic subfactors.