# What are all the possible indices for the finite depth subfactors?

Jones index theorem (1983) states that the set of all possible (finite) indices of subfactors is exactly $$\mathrm{Ind}=\{ 4 \cos(\pi/n)^2 \ | \ n \ge 3 \} \cup [4, \infty),$$ but if we restrict to the finite depth case, the set (say $$\mathrm{Ind_{fd}}$$) must become countable, because then the index is the squared norm of the principal graph (which then is a finite bipartite graph). According to this paper on page 63 (Afzaly-Morrison-Penneys, to appear in MAMS) there are exactly $$8$$ possible such indices in the interval $$(4,5.25]$$.

The set $$\mathrm{Ind_{fd}}$$ is multiplicative (i.e. $$\alpha, \beta \in \mathrm{Ind_{fd}} \Rightarrow \alpha \beta \in \mathrm{Ind_{fd}}$$) because the tensor product keeps the finite depth. Now, by this paper (Wassermann, 1998), for all $$m, there is a (finite depth) Jones-Wassermann subfactor of index $$\frac{\sin^2(n\pi/m)}{\sin^2(\pi/m)}$$, so that the set $$\mathrm{Ind_{fd}}$$ has an accumultation point at $$\alpha n^2$$ for all $$n \in \mathbb{N}_{\ge 2}$$ and $$\alpha \in \mathrm{Ind_{fd}}$$.

By this paper (Etingof-Nikshych-Ostrik, 2005), $$\mathrm{Ind_{fd}}$$ is contained in the set of positive cyclotomic integers (i.e. a positive elements of $$\mathbb{Z}[c_n]$$ with $$c_n = 2\cos(\pi/n)$$), which is a breakthrough in the understanding of $$\mathrm{Ind_{fd}}$$.

Now, I feel like that we can get even better, because by Theorem 3.2 in this paper (Bisch, 1994): $$\alpha \in \mathrm{Ind_{fd}} \setminus \mathbb{N} \Rightarrow \alpha^{-1} \mathbb{N} \cap \mathrm{Ind_{fd}} = \emptyset.$$ Example: $$n(3-\sqrt{5}) \not \in \mathrm{Ind_{fd}}$$ because $$2\frac{3+\sqrt{5}}{2} \cdot n(3-\sqrt{5}) = 4n$$, and $$\frac{3+\sqrt{5}}{2} = 4\cos^2(\pi/5)$$.

More strongly (see this comment) by extending Bisch's result to the ring of integers $$R=\mathcal{O}_K$$ of any cyclotomic number field $$K$$ (for example $$R=\mathbb{Z}[c_n]$$, but it can be something else): $$\alpha \in \mathrm{Ind_{fd}} \setminus R \Rightarrow \alpha^{-1} R \cap \mathrm{Ind_{fd}} = \emptyset.$$

The purpose of this post is not really to ask what exactly is $$\mathrm{Ind_{fd}}$$ (which seems unreachable), but new results about it, in particular inspired by above observations.