Let $(N\_1 \subset M\_1)$ and $(N\_2 \subset M\_2)$ be two **[maximal](https://mathoverflow.net/questions/133651/is-there-a-maximal-finite-depth-infinite-index-irreducible-subfactor)** subfactors. 


Their tensor product, the subfactor $(N\_1 \otimes N\_2 \subset M\_1 \otimes M\_2)$, admits four obvious intermediate subfactors : $N\_1 \otimes N\_2$, $M\_1 \otimes M\_2$, $N\_1 \otimes M\_2$ and $M\_1 \otimes N\_2$, but what about the **non-obvious** ? 


For example, the tensor product of $(R^{\mathbb{Z}/p\mathbb{Z}} \subset R)$ and $(R^{\mathbb{Z}/q\mathbb{Z}} \subset R)$, with $p$ and $q$ prime numbers, admits (at least) one **non-obvious** intermediate subfactor if and only if $p=q$. 


So in general we could speculate that there is (at least) one non-obvious intermediate subfactor if and only if the initial subfactors are isomorphic, but it's **false** if they are 2-supertransitive and of index > 2, because in this case, even if they are isomorphic, there is no room for a single one non-obvious.  

(see [Watatani](http://www.sciencedirect.com/science/article/pii/S002212369690110X) (1996) prop5.1 p329) 



> 
> What are the possible cases about **non-obvious** for the tensor
>  product of two maximal subfactors ?
> 
> 
> 




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**Remark** : the intermediate subfactor lattices $\mathcal{L}(R^{\mathbb{Z}/p\mathbb{Z}} \otimes R^{\mathbb{Z}/p\mathbb{Z}} \subset R \otimes R)$ are different for each $p$.  

**Theorem** ([Lukacs-Palfy 1986](http://link.springer.com/article/10.1007/BF01197131)): Let $G$ be a finite abelian group, $H$ a group. If the subgroups lattice of $G \times G$ and $H \times H$ are isomorphic then $G \simeq H$.



What are the intermediate subfactors of the tensor product of two maximal subfactors?
This answer came after a discussion with [Feng Xu](http://www.math.ucr.edu/home/Faculty/Xu.htm). The following more general result is true: 


*Theorem*:
Let $(N\_i \subset M\_i)$, $i=1,2$, be irreducible finite index subfactors. Then $$ \mathcal{L}(N\_1 \subset M\_1) \times \mathcal{L}(N\_2 \subset M\_2) \subsetneq \mathcal{L}(N\_1 \otimes N\_2 \subset M\_1 \otimes M\_2)$$ if and only if they are intermediate subfactors $N\_i \subseteq P\_i \subset Q\_i \subseteq M\_i$, $i=1,2$, such that $(P\_i \subset Q\_i)$ is depth $2$ and isomorphic to $(R^{\mathbb{A}\_i} \subset R)$, with $\mathbb{A}\_2 \simeq \mathbb{A}\_1^{cop}$ which is the Kac algebra $\mathbb{A}\_1$ with the opposite coproduct. 


*Proof:*
This theorem was proved in the $2$-supertransitive case by Y. Watatani. The general case was conjectured by the OP, and proved by a discussion with Feng Xu as follows:  

Let the intermediate subfactors $$N\_1 \otimes N\_2 \subseteq P\_1 \otimes P\_2 \subset R \subset Q\_1 \otimes Q\_2 \subseteq M\_1 \otimes M\_2$$ with $R$ not of tensor product form, $P\_1 \otimes P\_2$ and $Q\_1 \otimes Q\_2$ the closest (below and above resp.) to $R$ among them of tensor product form. Now using Proposition 3.5 (2) of [[xu]](http://www.sciencedirect.com/science/article/pii/S0001870813002351#), $(P\_i \subseteq Q\_i)$, $i=1,2$, are depth $2$, there corresponding Kac algebras, $\mathbb{A}\_i$, $i=1,2$, are very simple and $\mathbb{A}\_2 \simeq \mathbb{A}\_1^{cop}$ (see Definition 3.6 and Proposition 3.10 of [[xu]](http://www.sciencedirect.com/science/article/pii/S0001870813002351#)). The converse is given by Theorem 3.14 (2) of [[xu]](http://www.sciencedirect.com/science/article/pii/S0001870813002351#).


