A fusion category is called noncommutative if its Grothendieck ring is noncommutative. Let us call a fusion category strongly noncommutative if every fusion category Morita equivalent to it (i.e. same Drinfeld center up to equiv.) is noncommutative.
Question: Is there a strongly noncommutative fusion category (say over $\mathbb{C}$)?
If so, what are the known examples?
Note that if $G$ is a finite group then $Vec(G)$ is not strongly noncommutative (even if $G$ is noncommutative) because it is Morita equivalent to $Rep(G)$ which has a commutative Grothendieck ring. Moreover, the Extended Haagerup fusion categories are also not strongly noncommutative because they form a Morita equivalent class and one of them is commutative.
This post is in the same spirit than this one about strongly simple fusion categories. The main difference is that I know examples of strongly simple fusion categories whereas I do not know a single strongly noncommutative fusion category.
The next step would be about fusion categories which are both strongly simple and strongly noncommutative.
Best Answer
Consider the symmetric group group $G = S_3$ of order $6$. Then $\mathrm{H}^3_{\mathrm{gp}}(G;\mathrm{U}(1)) \cong \mathbb Z/6\mathbb Z$. Choose a generator $\omega \in \mathrm{H}^3_{\mathrm{gp}}(G;\mathrm{U}(1))$. Then $\omega$ restricts nontrivially to every nontrivial subgroup of $G$.
It follows that $\mathbf{Vec}^\omega[G]$ is not Morita-equivalent to any other fusion category. (Recall that, for any $G,\omega$, fusion categories equivalent to $\mathbf{Vec}^\omega[G]$ are indexed by pairs consisting of a subgroup $H \subset G$ together with a 2-cochain $\psi$ on $H$ solving $\mathrm{d}\psi = \omega|_H$.)
But $G$ is noncommutative.