Does unitarity and modularity constrain fusion multiplicities to be 0,1?

Question

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

Answer 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

Answer 2

I'll add an extra observation. Perhaps the reason why you are asking this question is because you are reading a paper which seems to be implicitly assuming that all fusion multiplicities are $0$ or $1$. Fusion categories in which all fusion multiplicities are $0$ or $1$ are known as multiplicity free.

The reason that these categories are special/important is because these are exactly the ones in which we can (simply) define $6j$ symbols, which are crucial tools in the numerical study of fusion categories. When quantifying fusion categories with numerical data it is hence often useful to restrict to this case.

Limiting one's view to multiplicity free modular tensor categories is not too restrictive however. The classification of Rowel-Stong-Wang shows that all MTCs of rank $\leq 4$ are multiplicity free: "ON CLASSIFICATION OF MODULAR TENSOR CATEGORIES". Even with higher ranks there are still lots of multiplicity free examples, and one should not have the impression that every sufficiently complicated MTC will have multiplicities. The general consensus in the community seems to be that every interesting phenominon of MTCs can be modeled by multiplicity free examples.