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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]

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  • $\begingroup$ 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? $\endgroup$ – Meths Jan 24 at 21:13
  • $\begingroup$ @Meths : What you expect is false for a non-commutative fusion ring because $n_{ij}^k = n_{j^*i^*}^{k^*}$. $\endgroup$ – Sebastien Palcoux Jan 24 at 22:16
  • $\begingroup$ About the ‘inverse’ axiom, note that there exist finite monoids which are not groups (see OEIS A058129). $\endgroup$ – Sebastien Palcoux Jan 24 at 23:01
  • $\begingroup$ 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. $\endgroup$ – Meths Jan 25 at 2:40
  • $\begingroup$ @Meths: Hint: consider the usual $*$-representation of $\mathcal{F}$ then $(b_ib_j)^*=b_{j^*}b_{i^*}$. $\endgroup$ – Sebastien Palcoux Jan 25 at 3:23
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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:

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

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