
> 
> All the [subfactors](http://en.wikipedia.org/wiki/Subfactor) here are **irreducible** inclusion of hyperfinite II$\_1$ factors.
> 
> 
> 


A subfactor $(N \subset M)$ is **Homogeneous Single Chain** ($HSC$) if its lattice of intermediate subfactors is a single chain: $N=P\_0 \subset P\_1 \subset \dots \subset P\_{n-1} \subset P\_n= M$, with $(P\_i \subset P\_{i+1}) \simeq (P\_0 \subset P\_1)$. 


We qualify such a subfactor by the **notation** $HSC^n\_{(P\_0 \subset P\_1)}$, because it's a subfactor whose lattice is a single chain of length $n$ and whose intermediate maximal inclusions are all isomorphic to $(P\_0 \subset P\_1)$. 



> 
> **Question**: Is there a $HSC^n\_{(P \subset Q)}$ subfactor for all maximal subfactor $(P \subset Q)$ and for all $n>1$ ? 
> 
> 
> 


**Warning**: The subfactors given by the basic construction are not irreducible.




---


**Group-subgroup formulation**: 


**Definition** : Let $(A \subset B)$ and $(C \subset D)$ be inclusions of groups, then $(A \subset B) \sim (C \subset D)$ if $(A/A\_B \subset B/A\_B) \simeq (C/C\_D \subset D/C\_D)$, with $A\_B$ the [normal core](http://groupprops.wiki-site.com/index.php/Normal_core) of $A$ in $B$. 


**Remark** : In the finite group case, if $(A \subset B) \sim (C \subset D)$ then $(R^B \subset R^A) \simeq (R^D \subset R^C)$, but the converse is false, see [here](https://mathoverflow.net/questions/136021/is-there-a-purely-group-theoretic-reformulation-of-an-equivalence-of-subgroups) ($R$ is the hyperfinite II$\_1$ factor). 


**Definition**: An inclusion of groups is $HSC^n\_{(A \subset B)}$ if its lattice of intermediate subgroups is a single chain of length $n$ and if its intermediate maximal inclusions of groups are all $\sim$ to $(A \subset B)$. 



> 
> **Question**: Is there a $HSC^n\_{(A \subset B)}$ inclusion for all maximal inclusion $(A \subset B)$ and for all $n>1$ ?
> 
> 
> 


**Remark**: An inclusion of group $(A \subset B)$ is given (up to $\sim$) by a transitive permutation group $G$ of degree $d = [A:B]$ (i.e. $G$ is the image of the coset representation and $(A \subset B) \sim (G\_1 \subset G$)).  

Idem a maximal inclusion of groups is given (up to $\sim$) by a primitive permutation group. 


**Notation**: A transitive group $G$ is $HSC^n\_{M}$ (with $M$ a primitive group) if $(G\_1 \subset G)$ is $HSC^n\_{(M\_1 \subset M)}$. 


**Experimental answer**: **Yes**, if we restrict to the transitive groups of degree $\le 30$.  

More precisely, let $G\_{d,r}$ be the $r$-th transitive group of degree $d$ given by the [GAP Data Library](http://www.gap-system.org/Datalib/trans.html) : 



```
gap> TransitiveGroup(5,4);  
A5 

```

**Remark**: $G\_{d,1}$ is the cyclic group $\mathbb{Z}/d\mathbb{Z}$ 


The primitive groups of deg. $\le 5$: $G\_{2,1}$ ; $G\_{3,1}$, $G\_{3,2}$ ; $G\_{4,4}$, $G\_{4,5}$ ; $G\_{5,1}$, $G\_{5,2}$,$G\_{5,3}$, $G\_{5,4}$, $G\_{5,5}$. 


The following tables gives the number of $HSC^n\_{G\_{d,r}}$ transitive groups of degree $\le 30$ related to the primitive group $G\_{d,r}$, with $n>1$, and the first examples. 


$\begin{array}{c|c}
 &G\_{2,1}&G\_{3,1}&G\_{3,2}&G\_{4,4}&G\_{4,5}&G\_{5,1}&G\_{5,2}&G\_{5,3}&G\_{5,4}&G\_{5,5} \newline 
 \hline
2 &2&4&9&18&30&8&15&31&1&5 \newline
 \hline
3 &15&152&476 \newline
 \hline
4 &597 
\end{array} $ 


$\begin{array}{c|c}
 &G\_{2,1}&G\_{3,1}&G\_{3,2}&G\_{4,4}&G\_{4,5}&G\_{5,1}&G\_{5,2}&G\_{5,3}&G\_{5,4}&G\_{5,5} \newline 
 \hline
2 &G\_{4,1}&G\_{9,1}&G\_{9,3}&G\_{16,63}&G\_{16,195}&G\_{25,1}&G\_{25,4}&G\_{25,8}&G\_{25,192}&G\_{25,196} \newline
 \hline
3 &G\_{8,1}&G\_{27,1}&G\_{27,8} \newline
 \hline
4 &G\_{16,1} 
\end{array} $ 


These tables show the existence of all the $HSC^n\_{G\_{d,r}}$ transitive groups of degree $\le 30$. 


**Remark**: There is one and only one $HSC^2\_{G\_{5,4}}$ transitive group: $G\_{25,192}$, of order $46656000000$, whereas $G\_{5,4} \simeq A\_5$ is order $60$. 



Existence of homogeneous single chain compositions of a given maximal subfactor?
**Yes** by free compositions:


**Theorem**: Let $N \subset M$ be an irreducible finite index subfactor with $P$ an intermediate subfactor ($N \subset P \subset M$) such that $N \subset M$ is a free composition of $N \subset P$ and $P \subset M$.  

If $L$ is another intermediate subfactor $N \subset L \subset M$, then $N \subset L \subset P$ or $P \subset L \subset M$. 


I've just written a proof in this [short note](https://drive.google.com/file/d/0B2P_JgZe-Zd0Wl81MnhUODBhM3c/edit?usp=sharing).  

This result was already in this [recent paper](http://arxiv.org/abs/1308.5656) of Zhengwei Liu
(thm 2.11 p 9) with a different proof.  

(Note that the two last paragraphs of my proof can be avoided by using Liu's thm 4.1 p18) 


Now, thanks to [this paper](http://www.sciencedirect.com/science/article/pii/S0022123600936073) of Sante Gnerre, there always exists a finite index irreducible subfactor realizing a free composition of two given finite index irreducible subfactors. 


**Conclusion**: Given a finite index irreducible maximal subfactor $(P\subset Q)$, there exists a $HSC\_{(P\subset Q)}^n$ subfactor for all $n$, it suffices to take free compositions of $(P\subset Q)$. 


**Remark**: A free compostion of group-subgroup subfactors is not group-subgroup in general, so it's still open at this level, and harder (see the last remark of the section "group-subgroup formulation" above). 


**Remark**: For non-free compositions, a sufficient condition for the theorem is true (i.e. "no extra intermediate") is : $\alpha \xi\_i \overline{\alpha}$ irreducible for all $i \neq 0$ (see notation [here](https://drive.google.com/file/d/0B2P_JgZe-Zd0Wl81MnhUODBhM3c/edit?usp=sharing)) [observation of Zhengwei and me].


**Remark-questions**: a free composition (with a non-trivial subfactor) is not a "finite depth" subfactor, so we can also ask, given a *finite depth* finite index irreducible maximal subfactor $(P\subset Q)$:  

- Is there a finite depth $HSC\_{(P\subset Q)}^n$ subfactor for all $n$? (or equivalently, a finite depth "quotient" of the free composition subfactor (planar algebra) with "no extra intermediate", see the remark above).  

- If $(P\subset Q)$ is depth $r$, what's the minimal depth of a $HSC\_{(P\subset Q)}^n$ subfactor?


