Let $\mathcal{C}$ be a **fusion category** and let $(H_1,...,H_r)$ be its simple objects.

$\mathcal{C}$ is **non-pointed** if at least one of its simple object has Perron-Frobenius dimension $ \neq 1$.

Let the **fusion rules**:

$$ H_i \boxtimes H_j \simeq \bigoplus_k M_{ij}^{k} \otimes H_k $$ with $M_{ij}^{k}$ the **multiplicity space** of finite dimension, then:
$$(H_i \boxtimes H_j) \boxtimes H_k \simeq \bigoplus_r M_{ij}^{r} \otimes (H_r \boxtimes H_k) \simeq \bigoplus_s (\bigoplus_r M_{ij}^{r} \otimes M_{rk}^{s}) \otimes H_s $$
$$H_i \boxtimes (H_j \boxtimes H_k) \simeq \bigoplus_r M_{jk}^{r} \otimes (H_i \boxtimes H_r) \simeq \bigoplus_s (\bigoplus_r M_{jk}^{r} \otimes M_{ir}^{s}) \otimes H_s $$

Let the **associativity isomorphisms**: $$a_{ijk}^{s} : \bigoplus_r M_{ij}^{r} \otimes M_{rk}^{s} \to \bigoplus_r M_{jk}^{r} \otimes M_{ir}^{s}$$

Finally, by choosing ordered bases, the $a_{ijk}^{s}$ have matrix form $A_{ijk}^{s}$, called the **associativity matrices**.

**Remark:** The associativity matrices check the **pentagonal equations**.

Question: Is there a non-pointed fusion category and a choice of ordered bases such that the associativity matrices are permutation matrices ?

**Remark:** the answer is obviously yes for the pointed fusion categories $\mathcal{C}_G$ with $H_g \boxtimes H_h \simeq H_{gh}$

for $g,h \in G$ a finite group, because we can choose the matrix $(1)$ for each associativity matrix.