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The paper Classification of metaplectic modular categories by Ardonne-Cheng-Rowell-Wang (2016) mentions (in Section 3) the Grothendieck ring for the metaplectic modular categories, i.e. $SO(N)_2$, $N>1$ odd (see below).

Now by exchanging $1$ and $Z$ in lines (2) and (3), we get a new family of fusion rings. 

Question 1: Are there (unitary) fusion categories corresponding to this new family?
(which could be called twisted metaplectic)

Question 2: Can this procedure be extended to other fusion categories?

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There is a way to make two new families: in the right hand-side of (2) and (3), put $$\oplus (X_1 \oplus X_2)^{\oplus \frac{N-1}{2}}$$ By this modification, you still have a fusion ring, but the objects $X_1, X_2$ are now of FPdim $N$. Again you have the usual and the twisted version. Note that after this modification, the usual case with $N=3$ corresponds to the Grothendieck ring of $\mathrm{Rep}(S_4)$.

Bonus question 1: Are there (unitary) fusion categories corresponding to these new families?

Bonus question 2: Can this procedure be extended to other fusion categories?

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In https://arxiv.org/pdf/2005.05544.pdf we describe a general procedure that accomplishes (1) and (2) which we call zesting.

In this case, let $\mathcal{C}$ be an odd metaplectic category. In the notation above, let $\lambda\in Z^2(\mathbb{Z}/2,Inv(\mathcal{C}))$ be chosen with $\lambda(1,1)=Z$. Then a new tensor product defined by $U\hat{\otimes} V\cong U\otimes V\otimes \lambda(d(U),d(V))$ where $d$ is the grading degree function, utilizing the $\mathbb{Z}/2$ grading on $\mathcal{C}$. In particular $X_1\hat{\otimes} X_2\cong X_1\otimes X_2\otimes Z\cong X_1\otimes X_1$ etc. As $Z$ is a boson (we are using the braiding on $\mathcal{C}$ in our construction) Proposition 6.3 in loc. cit. says that we further twist the associativity by a $3$-cocycle.
Proposition 6.4 shows that the resulting fusion category does not admit a braiding.

One can proceed in a similar way for general $N$-metaplectic categories too (i.e. $N$ can be odd or even).

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  • $\begingroup$ Thanks for your answer, this is exactly what I was looking for! Do you know if this procedure keeps the unitarity? Have you heard about the other procedure mentioned in second part of the post (i.e. the bonus questions)? (I just fixed a typo there, it was not (1) and (2), but (2) and (3)) $\endgroup$ – Sebastien Palcoux Aug 14 at 19:45
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    $\begingroup$ I believe that zesting does not ruin unitarity--the various morphisms that get changed are in the original category, just twisted by some roots of unity factors typically. While this is not a proof, I think one could make it rigorous. The other fusion rules could potentially be obtained as a Z2-equivariantization of the near-group categories of type Z/N +(N-1) in the Evans-Gannon notation. Just a guess, but the numerology seems to work out. $\endgroup$ – Eric Rowell Aug 19 at 14:47

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