MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.