# What are the first non-maximal non-group-subgroup simple irreducible subfactors?

Definition: For an irreducible (finite index) subfactor $(\mathcal{N} \subset \mathcal{M})$, an intermediate $(\mathcal{N} \subset \mathcal{P} \subset \mathcal{M})$
is normal if the biprojections $e_{\mathcal{P}}$ and $e_{\mathcal{P}'}$ are central in the $2$-boxes spaces of $(\mathcal{N} \subset \mathcal{M})$ and $(\mathcal{M}' \subset \mathcal{N}')$. Note that if these $2$-boxes spaces are abelian, then every intermediate is normal.

Remark: because Teruya's definition of normal uses $(\mathcal{M} \subset \mathcal{M}_1)$ instead of $(\mathcal{M}' \subset \mathcal{N}')$, for more caution (but I believe it's not necessary), the factors are supposed to be isomorphic to the hyperfinite ${\rm II}_1$ factor, or at least are supposed to be anti-isomorphic to themselves (see this paper and this post).

Examples: for $(\mathcal{N} \subset \mathcal{P} \subset \mathcal{M})$ with $(\mathcal{N} \subset \mathcal{M})$ isomorphic to the tensor or the free composition of $(\mathcal{N} \subset \mathcal{P})$ and $(\mathcal{P} \subset \mathcal{M})$, then $\mathcal{P}$ is a normal intermediate. For the group-subgroup subfactor of $(H \subset G)$, the normal intermediate subfactors are given by the intermediate subgroups $K$ with $HgK=KgH$, $\forall g \in G$. For a group subfactor, they are then given the normal subgroups.

Definition: An irreducible subfactor $(\mathcal{N} \subset \mathcal{M})$ is simple if it admits no non-trivial normal intermediate.

Examples: The maximal subfactors are simple. A group subfactor is simple iff the group is simple.
The first non-trivial non-maximal group-subgroup simple subfactor are indices $20$, $20$, $21$, $28$, given by:
$(\mathbb{Z}_3 \subset A_5)$, $(S_3 \subset S_5)$, $(D_8 \subset G)$ and $(S_3 \subset G)$ with $G = PSL(2,7)$.

Question: What are the first non-maximal non-group-subgroup simple irreducible subfactors?
(also non-Kac-coideal, because $(\mathbb{Z}_3 \subset A_5)$ could admit a simple non-trivial Kac deformation)

Remark: for the irreducible subfactors of index $<6$, the depth 1-2 edges of the principal graphs (and dual) are multiplicity one, so the $2$-boxes spaces are abelian, so all the biprojections are central, and so all the intermediates are normal; it follows that they can't be simple and non-maximal.

• what you mean by "first"? – Dave Penneys Sep 1 '14 at 19:21
• @DavePenneys: the "first" such subfactors regarding to the index. Nevertheless the first discovered or just an example, would be still interesting. – Sebastien Palcoux Sep 1 '14 at 19:25
• Whoops - I answered before I saw "non-maximal," so I deleted my answer. Have you checked the affine $A$'s yet? They are the first possible examples. – Dave Penneys Sep 1 '14 at 19:52
• @DavePenneys: For every irreducible subfactors of index $< 6$, the depth $1$-$2$ edges of the principal graph (and dual) are multiplicity one, so the $2$-boxes spaces are abelian, so all the biprojections are central, so all the intermediates are normal; it follows that they can't be simple and non-maximal. – Sebastien Palcoux Sep 2 '14 at 10:48
• – Sebastien Palcoux Sep 2 '14 at 10:48