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Theorem 3.2 in this paper (Bisch, 1994) states that for a finite depth ${\rm II}_1$ subfactor of integral index, every intermediate subfactor are also of integral index. As an application, every such subfactor of prime index must be maximal, moreover, many possible indices of such subfactors can be ruled out as follows: let $\alpha_1, \dots , \alpha_n > 0$ such that $\prod_i \alpha_i \in \mathbb{N}$, then one among the $\alpha_j \not \in \mathbb{N}$ cannot be the index of such a subfactor (proof: if all of them are, you get a contradiction with Bisch's theorem by considering a direct product of subfactors). For example, there is no such subfactor of index $n^{p/q} \not \in \mathbb{N}$ (where $n, p, q \in \mathbb{N}^{*})$, or there is none of index $n(3-\sqrt{5})$, because there are ones of indices $2$ and $\phi =(3+\sqrt{5})/2$, but $2 \cdot \phi \cdot n(3-\sqrt{5}) = 4n \in \mathbb{N}$. So we can a lot retrict the set of possible such indices like that (I don't know how much).

Now there is an other way to make such restrictions coming from fusion categories, using a theorem in this paper (Etingof-Nikshych-Ostrik, 2005), which implies that the index of such subfactors must be a cyclotomic integer, i.e. an element of $\mathbb{Z}[c_d]$ for some integer $d$, with $c_d=2\cos(2\pi/d)$.

Question: Can Bisch's theorem be extended to cyclotomic integers of fixed degree (i.e. if the index of a finite depth ${\rm II}_1$ subfactor is in $\mathbb{Z}[c_d]$, then so is for the index of the intermediate subfactors)?

As an application, we could get even more restrictions on possible indices of such subfactors, moreover, if such a subfactor is of index $\alpha \in \mathbb{Z}[c_d]$ and if $\beta$ is the index of an intermediate subfactor then $\beta\mathbb{Z}[c_d]$ would be a divisor ideal of $\alpha\mathbb{Z}[c_d]$; in particular, if $\alpha\mathbb{Z}[c_d]$ is a prime ideal, then the index of intermediate subfactors would only be units of $\mathbb{Z}[c_d]$ or units multiple of $\alpha$.

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    $\begingroup$ At first glance it looks like Bisch's argument applies without modification to the ring of integers in any field. Am I missing something? $\endgroup$
    – Noah Snyder
    Jul 3, 2020 at 23:51
  • $\begingroup$ @NoahSnyder: He used the following which should not hold for non-integral index subfactor: << We prove a simple criterion for finite depth of subfactors $N$ of a ${\rm II}_1$ factor $M$ with integer index. Such subfactors arise for instance from spin- and vertex model commuting squares >>. $\endgroup$
    – Sebastien Palcoux
    Jul 4, 2020 at 0:12
  • $\begingroup$ The sentence you quote isn't part of the proof. $\endgroup$
    – Noah Snyder
    Jul 4, 2020 at 0:48
  • $\begingroup$ @NoahSnyder: Yes. Then a stronger statement should hold (but perhaps more difficult to apply): the index of the intermediates of such a subfactor of index $α$ must be in the ring of integers of $\mathbb{Q}(α)$. Why Dietmar did not write such a fundamental result in his paper if its proof is is exactly the same as for the rational integers? $\endgroup$
    – Sebastien Palcoux
    Jul 4, 2020 at 1:32
  • $\begingroup$ Because it's a paper about when indices can be rational non-integers? $\endgroup$
    – Noah Snyder
    Jul 4, 2020 at 1:46

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