# What is the difference between the fusion rules for anyons and the decomposition of representations into the irreducible representations?

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I am studying topological quantum computation. In many references, it starts with the anyonic statistics. In my understanding, a type of anyon corresponds to an irreducible representation of the braid group, is it correct? Also,  fusion rules for composite anyons are discussed. There is my main question. Fusion rules are described as

$\phi_{a} \times \phi_{b} = \sum_c N^{a,b}_c \phi_{c},$

where a, b, and, c are types of anyons. My question is whether this formula is nothing but the decomposition of the product of two irreps for the braid group into direct sums of irreps, or not. If the answer is yes, I wonder why people call it fusion rule other than just the decomposition of the representation. If the answer is no, additional questions are following;

what is the difference between them?

why the category theory is needed to describe this rule?

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