# Is there a noncommutative simple fusion ring?

A fusion ring $$\mathcal{F}$$ is given by a finite set $$B = \{b_1,b_2, \dots, b_r \}$$ such that $$b_i b_j = \sum_k n_{i,j}^k b_k$$ with $$n_{i,j}^k \in \mathbb{Z}_{\ge 0}$$, satisfying axioms slightly augmenting the group axioms (see the details here).

The fusion ring $$\mathcal{F}$$ is called noncommutative if $$\exists i,j$$ with $$b_ib_j\neq b_jb_i$$. A fusion subring is given by a subset $$B_S=\{b_s \ | \ s \in S\} \subseteq B$$ with $$1 \in S = S^*$$ and $$\forall i,j \in S$$ then $$n_{i,j}^k \neq 0$$ only if $$k \in S$$. It is called simple when every fusion subring is given by the subset $$\{b_1\}$$ or $$B$$. Let $$G$$ be a finite group, then the Grothendieck ring of $$Rep(G)$$ is simple (as fusion ring) if and only if $$G$$ is simple.

Question: Is there a noncommutative simple fusion ring?

Investigation

According to this post, a noncommutative fusion ring must be of rank at least $$5$$, and at rank exactly $$5$$, we can assume: $$2^*=3$$, $$4^*=4$$ and $$5^*=5$$. A brute-force computation shows that at multiplicity at most three, there is a unique simple example (upto equiv.):
$$\begin{smallmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\end{smallmatrix} , \ \begin{smallmatrix}0&1&0&0&0\\0&0&1&1&0\\1&0&0&0&1\\0&0&1&0&1\\0&1&0&1&1\end{smallmatrix} , \ \begin{smallmatrix}0&0&1&0&0\\1&0&0&0&1\\0&1&0&1&0\\0&1&0&0&1\\0&0&1&1&1\end{smallmatrix} , \ \begin{smallmatrix}0&0&0&1&0\\0&0&1&0&1\\0&1&0&0&1\\1&0&0&1&1\\0&1&1&1&1\end{smallmatrix} , \ \begin{smallmatrix}0&0&0&0&1\\0&1&0&1&1\\0&0&1&1&1\\0&1&1&1&1\\1&1&1&1&2\end{smallmatrix}$$ but it is commutative. Note that $$[d(b_1),d(b_2),d(b_3),d(b_4),d(b_5)] =[1,\alpha,\alpha,\beta,\gamma]$$ with:

• $$\alpha =1+2cos(2\pi/7) \simeq 2.246979603\dots$$,
• $$\beta =1-2cos(6\pi/7) \simeq 2.801937735\dots$$,
• $$\gamma =\alpha+\beta-1 \simeq 4.048917339\dots$$,
• so that: $$gdim:=\sum_i d(b_i)^2 \simeq 36.65039990\dots$$,

moreover, it is of Frobenius type and satisfies Schur Product Property.

The fusion ring of one of the even parts of Extended Haagerup will work (see Appendix A of our paper). It has rank 8, and you can see that $$AB \neq BA$$. Surely there's less complicated examples though.
According to this paper (by Grossman-Morrison-Penneys-Peters-Snyder) the Morita equivalence class of the Extended Haagerup fusion category EH2 (the one mentioned in Noah's answer) contains exactly four fusion categories denoted EH1, EH2, EH3, EH4 (see the explicit fusion rules below). They are all simple of rank in $$\{6,8\}$$ and multiplicity in $$\{4,7\}$$, and the (three) ones of rank $$8$$ are also noncommutative (with Grothendieck ring generating the $$\mathbb{C}$$-algebra $$\mathbb{C}^4 \oplus M_2(\mathbb{C}))$$.