# A twisted Haagerup category without pivotal structure

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

: Izumi, Masaki The classification of $$3^n$$ subfactors and related fusion categories. Quantum Topol. 9 (2018), no. 3, 473–562.
: Ostrik, Victor. Pivotal fusion categories of rank 3. Mosc. Math. J. 15 (2015), no. 2, 373-396, 405.
: Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor. On fusion categories. Ann. of Math. (2) 162 (2005), no. 2, 581-642.

Andrew Schopieray pointed out to me the PhD thesis of Josiah E. Thornton .

Let $$\mathcal{F}$$ be a fusion ring with basis $$B=\{ b_1, \dots, b_r \}$$. Let $$G$$ be the group of invertible elements $$b_i$$ of $$B$$ (i.e. $$\textrm{FPdim}(b_i)=1$$). The fusion ring $$\mathcal{F}$$ is called quadratic if the action of $$G$$ on $$B \setminus G$$ is transitive. If $$G=B$$ such a fusion ring is called pointed, and if $$|B \setminus G|=1$$ it is called near-group. A fusion category with a quadratic Grothendieck ring is called (in the literature) a quadratic category or a generalized near-group category.

Theorem. A categorification $$\mathcal{C}$$ of a quadratic fusion ring must admit a spherical (so pivotal) structure.
Proof. By [2, Theorem IV.3.6.], $$\mathcal{C}$$ must be $$\varphi$$-pseudo-unitary (i.e. pseudo-unitary up to Galois automorphism), and then spherical (so pivotal) by [1, Proposition 2.16].

References

 V. Drinfeld, S. Gelaki, D. Nikshych, V. Ostrik, On braided fusion categories. I., Selecta Math. (N.S.) 16 (2010), no. 1, 1--119.

 J.E. Thornton, Generalized near-group categories, Thesis (Ph.D.)–University of Oregon (2012) 72 pp.