Let $G$ be a finite group, $\tau$ a group automorphism of $G$ of period two and $m$ a natural number. Following [1, Definition 2.1], a complex fusion category $\mathcal{C}$ is called a *quadratic category* with $(G,\tau,m)$ if its Grothendieck ring has basis $\{X_g, Y_g \ | \ g \in G\}$ and fusion rules (let $e$ be the neutral element of $G$):

- $X_gX_h = X_{gh}$,
- $X_gY_e = Y_g = Y_eX_{g^{\tau}}$,
- $Y_e^2 = X_e + m\sum_{g \in G} Y_g$.

Note that *quadratic* just requires the action of $G$ on the non-group part of the basis to be transitive, as stated in the second line above, in particular, we can have $Y_{g_1}=Y_{g_2}$ with $g_1 \neq g_2$.

Let $\mathcal{C}$ be a spherical quadratic category with $(G,\tau,m)$. By [1, Theorem 2.2], if $G$ is an odd group and $m$ is an odd number, then $G$ is abelian and $g^{\tau} = g^{-1}$ for any $g \in G$.

The Haagerup category is a quadratic category with $(C_3, -1, 1)$.

**Question**: Is there a complex fusion category, quadratic with $(C_3, 1, 1)$?

According to above statements, such a fusion category would be a *twisted* Haagerup category without spherical structure, in fact without pivotal structure by applying [2, Corollary 2.14].

Recall that the existence of a pivotal structure on every fusion category is a well-known conjecture [3, Conjecture 2.8]. If this conjecture is true then above question has a negative answer, but if this conjecture is false then above category could be a first counter-example. Anyway, it should be possible to answer above question independently of this conjecture.

*References*

[1]: Izumi, Masaki The classification of $3^n$ subfactors and related fusion categories. Quantum Topol. 9 (2018), no. 3, 473–562.

[2]: Ostrik, Victor. Pivotal fusion categories of rank 3. Mosc. Math. J. 15 (2015), no. 2, 373-396, 405.

[3]: Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor. On fusion categories. Ann. of Math. (2) 162 (2005), no. 2, 581-642.