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The Temperley-Lieb-Jones subfactor planar algebra $\mathcal{TLJ}_{\delta}$ admits the following properties:

  • maximal,
  • it exists for every possible index, i.e. $\delta^2 \in \{4cos^2(\pi/n) \ | \ n \le 3\} \cup [4, \infty]$,
  • for any subfactor planar algebra $\mathcal{P}$ of index $\delta^2$, there is a unital embedding $\mathcal{TLJ}_{\delta} \hookrightarrow_u \mathcal{P}$.
    ($u$ for unital)

It is the only (family of irreducible) subfactor planar algebra(s) to satisfy all the above properties.

The inclusion $S_{n-1} \subset S_n$ is a kind of group-theoretical analogous because of its properties:

  • (core-free) maximal,
  • it exists for every possible index, i.e. $n \in \mathbb{N}_{\ge 1}$,

let $\mathcal{PS}_n$ be the (irreducible) group-subgroup subfactor planar algebra $\mathcal{P}(S_{n-1} \subset S_n)$,

  • $\mathcal{PS}_n$ is isomorphic to $\mathcal{TLJ}_{n^{1/2}}$ for $n \le 3$.

It is the only (family of) core-free inclusion(s) of groups to satisfy the above three properties.

Let $G$ be a finite group and $H$ a (core-free) subgroup of index $n$.
Question: Is there a unital embedding $\mathcal{PS}_n \hookrightarrow_u \mathcal{P}(H \subset G)$?
If so, what about the converse?

More generally, let $G_i$ be a finite group and $H_i$ a (core-free) subgroup of index $n$ with $i=1,2$.
Extended question: What is the group-theoretical reformulation for the following property? $$\mathcal{P}(H_1 \subset G_1) \hookrightarrow_u \mathcal{P}(H_2 \subset G_2).$$

Remark: the group-theoretical reformulation for isomorphism is known (MR1920326, pointed out in this answer of Dave Penneys), let call it subfactor equivalence of inclusions.

Let $\mathcal{PA}_n$ be the group-subgroup subfactor planar algebra $\mathcal{P}(A_{n-1} \subset A_n)$, and $G,H$ as above. Alternating question: Is it true that $\mathcal{PA}_n \hookrightarrow_u \mathcal{P}(H \subset G)$ iff $G \hookrightarrow A_n$?

General knowledge: A number $n$ is called insipid if the groups having a core-free maximal subgroup of index $n$ are exactly $A_n$ and $S_n$. There is an OEIS enter for these numbers: A102842. There are exactly $486$ insipid numbers less than $1000$. In fact, the insipid numbers are of density one, and the number of non-insipid numbers less than $r$ grows like $2r/\log r$ (MR661693, pointed out in this answer of @verret).

Let call an index $d=\delta^2$ insipid if $\mathcal{TLJ}_{\delta}$ is the unique irreducible maximal subfactor planar algebras of index $d$. The insipid indices less than $4$ form the following set:

$$ \{4cos^2(\pi/n) \ | \ n \ge 3, \ n \neq 12, \ n \not \equiv 2 (\textrm{mod}\ 4) \}.$$

Bonus question: What is the full set of insipid indices?

Remark: By this answer of Noah Snyder, an insipid index must be less than $9$, because of the existence of infinite depth irreducible maximal subfactor planar algebras coming from $SU(3)$ at any index above $9$.

Let call an index $d \ge 4$ finite-insipid if an irreducible maximal subfactor planar algebra of index $d$ must be of infinite depth.

I will write a question about the finite-insipid indices later, any suggestion is welcome.

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