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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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References:
[1] ; Bakalov, Lectures on tensor categories and modular functors. Lectures on tensor categories and modular functors. University Lecture Series, 21 (2001) · Zbl 0965.18002
[2] 10.2140/ant.2015.9.1793 · Zbl 1330.18008
[3] 10.2307/1969820 · Zbl 0050.39304
[4] 10.2307/1969702 · Zbl 0055.41704
[5] 10.1155/S1073792804131206 · Zbl 1063.18005
[6] 10.1090/surv/205 · Zbl 1365.18001
[7] 10.1007/s00220-011-1329-3 · Zbl 1236.46055
[8] 10.1016/j.aim.2013.12.014 · Zbl 1304.18017
[9] 10.1016/j.aim.2017.01.015 · Zbl 1362.81083
[10] ; Isaacs, Character theory of finite groups. Character theory of finite groups. Pure and Applied Mathematics, 69 (1976) · Zbl 0337.20005
[11] 10.1007/s002200000234 · Zbl 1032.46529
[12] 10.1142/S0129055X01000818 · Zbl 1033.46506
[13] ; Izumi, Proceedings of the 2014 Maui and 2015 Qinhuangdao conferences in honour of Vaughan F. R. Jones’ 60th birthday. Proceedings of the 2014 Maui and 2015 Qinhuangdao conferences in honour of Vaughan F. R. Jones’ 60th birthday. Proc. Centre Math. Appl. Austral. Nat. Univ., 46, 222 (2017) · Zbl 1386.46002
[14] 10.1023/A:1009949909889 · Zbl 0971.16018
[15] 10.1016/j.aim.2003.12.004 · Zbl 1100.16033
[16] 10.1016/S0022-4049(02)00248-7 · Zbl 1033.18003
[17] 10.1016/j.aim.2006.07.017 · Zbl 1138.16017
[18] 10.1090/conm/441/08500
[19] 10.1090/S0002-9947-07-04276-6 · Zbl 1141.16028
[20] 10.1007/978-1-4684-9458-7
[21] 10.1016/j.jalgebra.2011.02.002 · Zbl 1236.18012
[22] 10.2206/kyushujm.56.59 · Zbl 1007.57010
[23] 10.1006/jabr.1998.7558 · Zbl 0923.46052
[24] 10.1016/0040-9383(63)90012-0 · Zbl 0215.39903
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