# Is there an integral fusion ring which is not of Frobenius type?

Combinatorially, a fusion ring $$\mathcal{F}$$ is nothing but a finite set $$B=\{b_1, \dots, b_r\}$$ (generating the $$\mathbb{Z}$$-module $$\mathbb{Z} B$$) together with fusion rules: $$b_i \cdot b_j = \sum_{k=1}^r n_{i,j}^k b_k$$ with $$n_{i,j}^k \in \mathbb{Z}_{\ge 0}$$, satisfying axioms slightly augmenting the group axioms:

• (Associativity) $$b_i \cdot (b_j \cdot b_k) = (b_i \cdot b_j) \cdot b_k$$ , i.e., $$\sum_s n_{ij}^sn_{sk}^t = \sum_s n_{jk}^sn_{is}^t$$.
• (Neutral) $$b_1 \cdot b_i = b_i \cdot b_1 = b_i$$, i.e., $$n_{1i}^j = n_{i1}^j = \delta_{ij}$$.
• (Inverse/Adjoint) $$\forall i \ \exists!j$$ (denoted $$i^*$$) such that $$n_{ij}^1>0$$. In addition, $$n_{i^*,k}^{1} = n_{k,i^*}^{1} = \delta_{i,k}$$.
• Frobenius-Perron reciprocity: $$n_{ij}^k = n_{i^*k}^j = n_{kj^*}^i$$.

It follows that:

• $$*$$ induces an antihomomorphism of algebra, providing a structure of $$*$$-algebra to $$\mathbb{C}\mathcal{B}$$,
• Frobenius-Perron theorem: $$\exists!$$ $$*$$-homomorphism $$d:\mathbb{C}\mathcal{B} \to \mathbb{C}$$ with $$d(\mathcal{B}) \subset (0,\infty)$$, with $$\mathbb{C}\mathcal{B}$$ is a finite dimensional von Neumann algebra given by $$b_i^* = b_{i^*}$$.

The number $$d(b_i)$$ is called the Frobenius-Perron dimension of $$b_i$$, whereas $$\sum_i d(b_i)^2$$ is called the Frobenius-Perron of $$\mathcal{F}$$, noted $$\mathrm{FPdim}(\mathcal{F})$$. Let $$[d(b_1), d(b_2), \dots , d(b_r)]$$ be the type of $$\mathcal{F}$$.

The fusion ring $$\mathcal{F}$$ is called:

• of Frobenius type if for all $$i$$, $$\frac{\mathrm{FPdim}(\mathcal{F})}{d(b_i)}$$ is an algebraic integer,
• integral if for all $$i$$ the number $$d(b_i)$$ is an integer, and then Frobenius type just means that $$d(b_i)$$ divides $$\mathrm{FPdim}(\mathcal{F})$$ for all $$i$$,
• commutative if for all $$i,j$$, $$b_i \cdot b_j = b_j \cdot b_i$$, i.e., $$n_{i,j}^k = n_{j,i}^k$$.

It is a famous open problem whether the Grothendieck ring of a fusion category is of Frobenius type. Now, George Kac proved MR0304552 that the Grothendieck ring of $$Rep(K)$$ with $$K$$ a finite dimensional Kac algebra (i.e. Hopf $$*$$-algebra) is of Frobenius type. Now the fusion category $$Rep(K)$$ is unitary and integral.

Now, there are many fusion rings which are not Grothendieck rings of a fusion category, so that perhaps fusion rings which are not of Frobenius type are already known.
Consider the following three properties for a fusion ring:
(1) integral,
(2) commutative,
(3) unitary (i.e. admits a unitary categorification).

Consider a subset $$S \subseteq \{1,2,3\}$$, then:

Question ($$S$$): Is there a fusion ring satisfying (i) for all $$i \in S$$, but not of Frobenius type?
[it is a unified way to ask $$2^3=8$$ questions]

• Is existence of an inverse actually regarded as an axiom for a fusion ring? (See e.g. Definition 3.1.7 of Etingof et al.'s 'Tensor Categories'). Under the inverse assumption, Frobenius-Perron reciprocity easily follows by associativity - is it also possible to show that $n_{ij}^{k}=n_{i*j*}^{k*}$ by any chance? – Meths Jan 24 at 21:13
• @Meths : What you expect is false for a non-commutative fusion ring because $n_{ij}^k = n_{j^*i^*}^{k^*}$. – Sebastien Palcoux Jan 24 at 22:16
• About the ‘inverse’ axiom, note that there exist finite monoids which are not groups (see OEIS A058129). – Sebastien Palcoux Jan 24 at 23:01
• Any chance you can direct me to a reference for $n_{ij}^{k}=n_{j*i*}^{k*}$ (or give a hint as to how to get there)? Also, I don't quite see the relevance of monoids (unless we consider the monoid to consist of elements $\text{span}_{\mathbb{N}}(B)$ ,with multiplication amongst the elements defined in a way consistent with the above) - in which case these are infinite monoids anyway. – Meths Jan 25 at 2:40
• @Meths: Hint: consider the usual $*$-representation of $\mathcal{F}$ then $(b_ib_j)^*=b_{j^*}b_{i^*}$. – Sebastien Palcoux Jan 25 at 3:23

If $$3 \not \in S$$ then the answer to Question($$S$$) is yes.

There are integral commutative fusion rings which are not of Frobenius type.

Examples:

• Non-simple: rank $$4$$, FPdim $$15$$, type $$[1,1,2,3]$$, and fusion rules:

$$\begin{smallmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{smallmatrix} , \ \begin{smallmatrix} 0&1&0&0\\1&0&0&0\\0&0&1&0\\0&0&0&1 \end{smallmatrix} , \ \begin{smallmatrix} 0&0&1&0\\0&0&1&0\\1&1&1&0\\0&0&0&2 \end{smallmatrix} , \ \begin{smallmatrix} 0&0&0&1\\0&0&0&1\\0&0&0&2\\1&1&2&1 \end{smallmatrix}$$

• Simple: rank $$6$$, FPdim $$143$$, type $$[1,4,4,5,6,7]$$, and fusion rules:

$$\begin{smallmatrix}1&0&0&0&0&0\\0&1&0&0&0&0\\0&0&1&0&0&0\\0&0&0&1&0&0\\0&0&0&0&1&0\\0&0&0&0&0&1\end{smallmatrix} , \ \begin{smallmatrix}0&1&0&0&0&0\\1&0&1&1&1&0\\0&1&0&1&0&1\\0&1&1&1&0&1\\0&1&0&0&1&2\\0&0&1&1&2&1\end{smallmatrix} , \ \begin{smallmatrix}0&0&1&0&0&0\\0&1&0&1&0&1\\1&0&2&0&0&1\\0&1&0&2&1&0\\0&0&0&1&2&1\\0&1&1&0&1&2\end{smallmatrix} , \ \begin{smallmatrix}0&0&0&1&0&0\\0&1&1&1&0&1\\0&1&0&2&1&0\\1&1&2&1&0&1\\0&0&1&0&2&2\\0&1&0&1&2&2\end{smallmatrix} , \ \begin{smallmatrix}0&0&0&0&1&0\\0&1&0&0&1&2\\0&0&0&1&2&1\\0&0&1&0&2&2\\1&1&2&2&1&1\\0&2&1&2&1&2\end{smallmatrix} , \ \begin{smallmatrix}0&0&0&0&0&1\\0&0&1&1&2&1\\0&1&1&0&1&2\\0&1&0&1&2&2\\0&2&1&2&1&2\\1&1&2&2&2&2\end{smallmatrix}$$

Note that $$15= 3 \times 5$$ and $$143 = 11 \times 13$$. They admit no categorification because by MR2098028, any fusion category of Frobenius-Perron dimension $$pq$$ (with $$p,q$$ different odd primes) is group-theoretical, whereas by MR2735754, a (weakly) group theoretical fusion category is of Frobenius type.

Now, four new simple integral (commutative) fusion rings not of Frobenius type and on which a unitary categorification cannot be excluded according to my current knowledge:

• rank $$6$$, FPdim $$924 = 2^2 \cdot 3 \cdot 7 \cdot 11$$, type $$[1,7,8,12,15,21]$$ and fusion rules:

$$\begin{smallmatrix}1&0&0&0&0&0 \\ 0&1&0&0&0&0 \\ 0&0&1&0&0&0 \\ 0&0&0&1&0&0 \\ 0&0&0&0&1&0 \\ 0&0&0&0&0&1\end{smallmatrix} , \ \begin{smallmatrix}0&1&0&0&0&0 \\ 1&0&0&1&1&1 \\ 0&0&1&1&1&1 \\ 0&1&1&1&1&2 \\ 0&1&1&1&1&3 \\ 0&1&1&2&3&3\end{smallmatrix} , \ \begin{smallmatrix}0&0&1&0&0&0 \\ 0&0&1&1&1&1 \\ 1&1&1&1&1&1 \\ 0&1&1&2&1&2 \\ 0&1&1&1&2&3 \\ 0&1&1&2&3&4\end{smallmatrix} , \ \begin{smallmatrix}0&0&0&1&0&0 \\ 0&1&1&1&1&2 \\ 0&1&1&2&1&2 \\ 1&1&2&1&3&3 \\ 0&1&1&3&3&4 \\ 0&2&2&3&4&6\end{smallmatrix} , \ \begin{smallmatrix}0&0&0&0&1&0 \\ 0&1&1&1&1&3 \\ 0&1&1&1&2&3 \\ 0&1&1&3&3&4 \\ 1&1&2&3&4&5 \\ 0&3&3&4&5&7\end{smallmatrix} , \ \begin{smallmatrix}0&0&0&0&0&1 \\ 0&1&1&2&3&3 \\ 0&1&1&2&3&4 \\ 0&2&2&3&4&6 \\ 0&3&3&4&5&7 \\ 1&3&4&6&7&10\end{smallmatrix}$$

• rank $$6$$, FPdim $$1320 = 2^3 \cdot 3 \cdot 5 \cdot 11$$, type $$[1,9,10,11,21,24]$$ and fusion rules:

$$\begin{smallmatrix}1&0&0&0&0&0 \\ 0&1&0&0&0&0 \\ 0&0&1&0&0&0 \\ 0&0&0&1&0&0 \\ 0&0&0&0&1&0 \\ 0&0&0&0&0&1\end{smallmatrix} , \ \begin{smallmatrix}0&1&0&0&0&0 \\ 1&0&0&1&1&2 \\ 0&0&1&1&1&2 \\ 0&1&1&1&1&2 \\ 0&1&1&1&3&4 \\ 0&2&2&2&4&3\end{smallmatrix} , \ \begin{smallmatrix}0&0&1&0&0&0 \\ 0&0&1&1&1&2 \\ 1&1&0&0&2&2 \\ 0&1&0&1&2&2 \\ 0&1&2&2&3&4 \\ 0&2&2&2&4&4\end{smallmatrix} , \ \begin{smallmatrix}0&0&0&1&0&0 \\ 0&1&1&1&1&2 \\ 0&1&0&1&2&2 \\ 1&1&1&1&2&2 \\ 0&1&2&2&4&4 \\ 0&2&2&2&4&5\end{smallmatrix} , \ \begin{smallmatrix}0&0&0&0&1&0 \\ 0&1&1&1&3&4 \\ 0&1&2&2&3&4 \\ 0&1&2&2&4&4 \\ 1&3&3&4&7&8 \\ 0&4&4&4&8&9\end{smallmatrix} , \ \begin{smallmatrix}0&0&0&0&0&1 \\ 0&2&2&2&4&3 \\ 0&2&2&2&4&4 \\ 0&2&2&2&4&5 \\ 0&4&4&4&8&9 \\ 1&3&4&5&9&11\end{smallmatrix}$$

• rank $$7$$, FPdim $$560 = 2^4 \cdot 5 \cdot 7$$, type $$[1,6,7,7,10,10,15]$$ and fusion rules:

$$\begin{smallmatrix}1&0&0&0&0&0&0\\0&1&0&0&0&0&0\\0&0&1&0&0&0&0\\0&0&0&1&0&0&0\\0&0&0&0&1&0&0\\0&0&0&0&0&1&0\\0&0&0&0&0&0&1\end{smallmatrix} ,\begin{smallmatrix}0&1&0&0&0&0&0\\1&0&0&0&1&1&1\\0&0&1&0&1&1&1\\0&0&0&1&1&1&1\\0&1&1&1&0&1&2\\0&1&1&1&1&0&2\\0&1&1&1&2&2&2\end{smallmatrix}, \begin{smallmatrix}0&0&1&0&0&0&0\\0&0&1&0&1&1&1\\1&1&0&1&1&1&1\\0&0&1&1&1&1&1\\0&1&1&1&1&1&2\\0&1&1&1&1&1&2\\0&1&1&1&2&2&3\end{smallmatrix} , \begin{smallmatrix}0&0&0&1&0&0&0\\0&0&0&1&1&1&1\\0&0&1&1&1&1&1\\1&1&1&0&1&1&1\\0&1&1&1&1&1&2\\0&1&1&1&1&1&2\\0&1&1&1&2&2&3\end{smallmatrix} , \begin{smallmatrix}0&0&0&0&1&0&0\\0&1&1&1&0&1&2\\0&1&1&1&1&1&2\\0&1&1&1&1&1&2\\0&1&1&1&2&3&2\\1&0&1&1&2&2&3\\0&2&2&2&3&2&4\end{smallmatrix} , \begin{smallmatrix}0&0&0&0&0&1&0\\0&1&1&1&1&0&2\\0&1&1&1&1&1&2\\0&1&1&1&1&1&2\\1&0&1&1&2&2&3\\0&1&1&1&3&2&2\\0&2&2&2&2&3&4\end{smallmatrix} , \begin{smallmatrix}0&0&0&0&0&0&1\\0&1&1&1&2&2&2\\0&1&1&1&2&2&3\\0&1&1&1&2&2&3\\0&2&2&2&3&2&4\\0&2&2&2&2&3&4\\1&2&3&3&4&4&6\end{smallmatrix}$$

• rank $$7$$, FPdim $$798=2 \cdot 3 \cdot 7 \cdot 19$$, type $$[1,7,8,9,9,9,21]$$ and fusion rules:

$$\begin{smallmatrix}1&0&0&0&0&0&0\\0&1&0&0&0&0&0\\0&0&1&0&0&0&0\\0&0&0&1&0&0&0\\0&0&0&0&1&0&0\\0&0&0&0&0&1&0\\0&0&0&0&0&0&1\end{smallmatrix} , \begin{smallmatrix}0&1&0&0&0&0&0\\1&0&0&1&1&1&1\\0&0&1&1&1&1&1\\0&1&1&1&1&1&1\\0&1&1&1&1&1&1\\0&1&1&1&1&1&1\\0&1&1&1&1&1&5\end{smallmatrix} , \begin{smallmatrix}0&0&1&0&0&0&0\\0&0&1&1&1&1&1\\1&1&1&1&1&1&1\\0&1&1&2&1&1&1\\0&1&1&1&2&1&1\\0&1&1&1&1&2&1\\0&1&1&1&1&1&6\end{smallmatrix} , \begin{smallmatrix}0&0&0&1&0&0&0\\0&1&1&1&1&1&1\\0&1&1&2&1&1&1\\1&1&2&1&1&2&1\\0&1&1&1&2&2&1\\0&1&1&2&2&1&1\\0&1&1&1&1&1&7\end{smallmatrix} , \begin{smallmatrix}0&0&0&0&1&0&0\\0&1&1&1&1&1&1\\0&1&1&1&2&1&1\\0&1&1&1&2&2&1\\1&1&2&2&1&1&1\\0&1&1&2&1&2&1\\0&1&1&1&1&1&7\end{smallmatrix} , \begin{smallmatrix}0&0&0&0&0&1&0\\0&1&1&1&1&1&1\\0&1&1&1&1&2&1\\0&1&1&2&2&1&1\\0&1&1&2&1&2&1\\1&1&2&1&2&1&1\\0&1&1&1&1&1&7\end{smallmatrix} , \begin{smallmatrix}0&0&0&0&0&0&1\\0&1&1&1&1&1&5\\0&1&1&1&1&1&6\\0&1&1&1&1&1&7\\0&1&1&1&1&1&7\\0&1&1&1&1&1&7\\1&5&6&7&7&7&8\end{smallmatrix}$$