If I have a braided tensor category that's unitary and modular, then how does the unitarity and modularity constrain the fusion multiplicities?


I know that if $a,b,c \in ob({C})$ satisfy the fusion rule $a \otimes b = \oplus\_c N^c\_{ab} c $ then $N^c\_{ab} \in \mathbb{Z}\_+$ for all possible non-trivial fusion channels. Does being unitarity and modular guarantee that $N^c\_{ab} = 0,1$ for all fusion channels?


I require it specifically for the category Rep(D(G)), which I know to be unitary modular. If it's in general not true, but is for this specific category, I'd still be happy!


Edit: <https://arxiv.org/pdf/2306.05560.pdf> computes the fusion rules of Rep(D(G)) to have $N^c\_{ab} =0,1$ for all dihedral and dicyclic groups, which lends some credence to this hypothesis



Does unitarity and modularity constrain fusion multiplicities to be 0,1?
This is false for $D(G)$, when $G$ is sufficiently complicated. For a finite group $G$, the representation category of $D(G)$ has irreducible objects parametrized by pairs $(g, V)$ where $g$ is a conjugacy class representative in $G$ and $V$ is an irreducible representation of the centralizer of $g$. The monoidal structure for pairs of the form $(1,V)$ coincides with the tensor product of representations of $G$, so we can find a counterexample just by finding a finite group $G$ and a pair of irreducible representations whose tensor product has multiplicity greater than 1.


For example, when $G = A\_4$, we have $N\_{V,V}^V = 2$ when $V$ is the unique irreducible representation of dimension 3, i.e., $V \otimes V$ contains $V \oplus V$.


You can check this in GAP:



```
t := CharacterTable(AlternatingGroup(4));
Irr(t)[4];                                  # yields [3,-1,0,0]
ScalarProduct(t,Irr(t)[4],[9,1,0,0]);       # yields 2

```

