# Fusion categories with permutation “associativity matrices”

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.

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How is $A^s_{ijk}$ a matrix if it has 4 indices? – Turion Feb 4 '14 at 16:24
Oh, I see. You mean that each $A^s_{ijk}$ is a matrix. My bad. – Turion Feb 4 '14 at 16:31