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Frobenius-Schur indicators for near-group and Haagerup-Izumi fusion categories. (English) Zbl 1442.18041
The main results of the present paper concern the computation of the higher Frobenius-Schur indicators of certain spherical fusion categories and the consequent investigation of the FS indicator rigidity of their Grothendieck rings.
In what follows a fusion category is, broadly speaking, a category whose objects behave like finite-dimensional complex representations of a finite group: it is a semisimple locally finite $$\mathbb{C}$$-linear abelian rigid (strict) monoidal category $$(\mathcal{C},\otimes,\boldsymbol{1})$$ with finitely many irreducible objects and such that $$\mathsf{End}_{\mathcal{C}}(\boldsymbol{1}) \cong \mathbb{C}$$ and $$\boldsymbol{1}$$ is irreducible. A pivotal structure on a fusion category is a choice of a natural monoidal isomorphism $$(-)^{**}\cong (-)$$ from the double dual endofunctor to the identity endofunctor. This allows us to consider left and right pivotal (or quantum) traces. A pivotal fusion category is spherical if the pivotal traces coincide, giving rise to a well-defined notion of trace of a morphism. An important invariant of spherical fusion categories is provided by the (categorical) Frobenius-Schur indicators introduced in [S.-H. Ng and P. Schauenburg, Contemp. Math. 441, 63–90 (2007; Zbl 1153.18008)]. If $$n$$ is a positive integer and $$V$$ is an object in a spherical fusion category $$\mathcal{C}$$, then the $$n$$-th Frobenius-Schur indicator of $$V$$ is the pivotal trace of a certain endomorphism of the finite-dimensional vector space $$\mathsf{Hom}_{\mathcal{C}}(\boldsymbol{1},V^{\otimes n})$$.
The Grothendieck ring $$K_0(\mathcal{C})$$ of a fusion category $$\mathcal{C}$$ is the $$\mathbb{Z}$$-based ring with basis given by the set of isomorphism classes of simple objects in $$\mathcal{C}$$, addition given by the direct sum in $$\mathcal{C}$$ and multiplication by the tensor product. We say that $$\mathcal{C}$$ categorifies a ring $$K$$ if $$K_0(\mathcal{C}) \cong K$$ and that a ring $$K$$ exhibits FS indicator rigidity if its categorifications can all be distinguished by their Frobenius-Schur indicators.
Henceforth, let $$G$$ be a finite group. A fusion category $$\mathcal{C}$$ is a near-group if its Grothendieck ring is given by $$K_0(\mathcal{C}) = \mathbb{Z}\left[G \cup \{\rho\}\right]$$, where multiplication is given by the group law and $\rho g = \rho = g\rho \qquad \text{and} \qquad \rho^2 = m\rho + \sum_{h \in G} h$ for $$m \in \mathbb{N}$$ and for all $$g \in G$$. It is a Tambara-Yamamagami category if $$m = 0$$. It is a Haagerup-Izumi fusion category if $$K_0(\mathcal{C}) =\mathbb{Z}\left[G \cup \{g\rho \mid g \in G\}\right]$$, where multiplication is given by the group law and $g(h\rho) = (gh)\rho = (h\rho)g^{-1} \qquad \text{and} \qquad (g\rho)(h\rho) = gh^{-1} + \sum_{x \in G} x\rho$ for all $$g,h \in G$$. By following the notation introduced in the paper, we set $$\mathrm{NG}(G,m) := \mathbb{Z}\left[G \cup \{\rho\}\right]$$ and $$\mathrm{HI}(G) := \mathbb{Z}\left[G \cup \{g\rho \mid g \in G\}\right]$$.
It is known [T. Basak and R. Johnson, Algebra Number Theory 9, No. 8, 1793–1823 (2015; Zbl 1330.18008)] that the rings $$\mathrm{NG}(G,0)$$ exhibit FS indicator rigidity. In the present paper, by taking advantage of the classifications provided in [M. Izumi, Rev. Math. Phys. 13, No. 5, 603–674 (2001; Zbl 1033.46506); D. E. Evans and T. Gannon, Adv. Math. 255, 586–640 (2014; Zbl 1304.18017); Adv. Math. 310, 1-43 (2017; Zbl 1362.81083)] and of the formula for computing Frobenius-Schur indicators by resorting to modular data of the Drinfel’d center from [S.-H. Ng and P. Schauenburg, Adv. Math. 211, No. 1, 34–71 (2007; Zbl 1138.16017)], the author computes the indicators for the non-invertible object $$\rho$$ in the known cases, concluding that:
$$\mathbf{(1)}$$ The near-group fusion ring $$\mathrm{NG}(G,|G|-1)$$ exhibits FS indicator rigidity.
$$\mathbf{(2)}$$ In all known near-group categories with $$m = |G|$$, the non-invertible object has Frobenius-Schur indicators given by quadratic Gauss sums. Moreover, the near-group fusion ring $$\mathrm{NG}(G,|G|)$$ for $$G=\mathbb{Z}/13\mathbb{Z}$$ does not have FS indicator rigidity, but the lesser odd order groups do exhibit FS indicator rigidity.
$$\mathbf{(3)}$$ All known Haagerup-Izumi categories have Frobenius-Schur indicators given by quadratic Gauss sums. Moreover, the Haagerup-Izumi fusion rings do not have FS indicator rigidity.
##### MSC:
 18M20 Fusion categories, modular tensor categories, modular functors 16T05 Hopf algebras and their applications 46L37 Subfactors and their classification
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