Higher Frobenius-Schur Indicators for Pivotal Categories
Siu-Hung Ng, Peter Schauenburg
备注：
A paragraph which describes the organization of the paper has been added to the introduction. Some observations have been added to Theorems 5.1 and 7.6
We define higher Frobenius-Schur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a $k$-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a $k$-linear semisimple pivotal monoidal category -- where both notions are defined --, the Frobenius-Schur indicators can be computed as traces of the Frobenius-Schur endomorphisms.
We give an explicit description, up to gauge equivalence, of group-theoretical quasi-Hopf algebras. We use this description to compute the Frobenius-Schur indicators for group-theoretical fusion categories.
In this paper, we propose a new approach towards the classification of spherical fusion categories by their Frobenius-Schur exponents. We classify spherical fusion categories of Frobenius-Schur exponent 2 up to monoidal equivalence. We also classify modular categories of Frobenius-Schur exponent 2 up to braided monoidal equivalence. It turns out that the Gauss sum is a complete invariant for modular categories of Frobenius-Schur exponent 2. This result can be viewed as a categorical analog of Arf's theorem on the classification of non-degenerate quadratic forms over fields of characteristic 2.
Group-theoretical fusion categories are defined by data concerning finite groups and their cohomology: A finite group $G$ endowed with a three-cocycle $ω$, and a subgroup $H\subset G$ endowed with a two-cochain whose coboundary is the restriction of $ω$. The objects of the category are $G$-graded vector spaces with suitably twisted $H$-actions; the associativity of tensor products is controlled by $ω$. Simple objects are parametrized in terms of projective representations of finite groups, namely of the stabilizers in $H$ of right $H$-cosets in $G$, with respect to two-cocycles defined by the …
We study Frobenius-Schur indicators of the regular representations of finite-dimensional semisimple Hopf algebras, especially group-theoretical ones. Those of various Hopf algebras are computed explicitly. In view of our computational results, we formulate the theorem of Frobenius for semisimple Hopf algebras and give some partial results on this problem.
In this paper, we introduce the notion of the pivotal cover $\mathcal{C}^{\mathsf{piv}}$ of a left rigid monoidal category $\mathcal{C}$ to develop a theoretical foundation for the theory of Frobenius-Schur (FS) indicators in "non-pivotal" settings. For an object $\mathbf{V} \in \mathcal{C}^{\mathsf{piv}}$, the $(n, r)$-th FS indicator $ν_{n, r}(\mathbf{V})$ is defined by generalizing that of an object of a pivotal monoidal category. This notion gives a categorical viewpoint to some recent results on generalizations of FS indicators. Based on our framework, we also study the FS indicators of t…
The Turaev-Viro state sum invariant can be extended to 3-manifolds with free boundaries. We use this fact to describe generalized Frobenius-Schur indicators as Turaev-Viro invariants of solid tori. This provides a geometric understanding of the $\mathrm{SL}(2,\mathbb{Z})$-equivariance of these indicators.
We consider a subclass of the class of group-theoretical fusion categories: To every finite group $G$ and subgroup $H$ one can associate the category of $G$-graded vector spaces with a two-sided $H$-action compatible with the grading. We derive a formula that computes higher Frobenius-Schur indicators for the objects in such a category using the combinatorics and representation theory of the groups involved in their construction. We calculate some explicit examples for inclusions of symmetric groups.
We introduce formulae of Frobenius-Schur indicators of simple objects of Tambara-Yamagami categories. By using techniques of the Fourier transform on finite abelian groups, we study some arithmetic properties of indicators.
The quantum Frobenius map and it splitting are shown to descend to corresponding maps for generalized $q$-Schur algebras at a root of unity. We also define analogs of $q$-Schur algebras for any affine algebra, and prove the corresponding results for them.
Ng and Schauenburg generalized higher Frobenius-Schur indicators to pivotal fusion categories and showed that these indicators may be computed utilizing the modular data of the Drinfel'd center of the given category. We consider two classes of fusion categories generated by a single non-invertible simple object: near groups, those fusion categories with one non-invertible simple object, and Haagerup-Izumi categories, those with one non-invertible simple object for every invertible object. Examples of both types arise as representations of finite or quantum groups or as Jones standard invariant…
Mason and Ng have given a generalization to semisimple quasi-Hopf algebras of Linchenko and Montgomery's generalization to semisimple Hopf algebras of the classical Frobenius-Schur theorem for group representations. We give a simplified proof, in particular a somewhat conceptual derivation of the appropriate form of the Frobenius-Schur indicator that indicates if and in which of two possible fashions a given simple module is self-dual.
Let $U:\mathcal{C}\rightarrow\mathcal{D}$ be a strong monoidal functor between abelian monoidal categories admitting a right adjoint $R$, such that $R$ is exact, faithful and the adjunction $U\dashv R$ is coHopf. Building on the work of Balan, we show that $R$ is separable (resp., special) Frobenius monoidal if and only if $R(\mathbb{1}_{\mathcal{D}})$ is a separable (resp., special) Frobenius algebra in $\mathcal{C}$. If further, $\mathcal{C},\mathcal{D}$ are pivotal (resp., ribbon) categories and $U$ is a pivotal (resp., braided pivotal) functor, then $R$ is a pivotal (resp., ribbon) functor…
The classical Frobenius-Schur indicators for finite groups are character sums defined for any representation and any integer m greater or equal to 2. In the familiar case m=2, the Frobenius-Schur indicator partitions the irreducible representations over the complex numbers into real, complex, and quaternionic representations. In recent years, several generalizations of these invariants have been introduced. Bump and Ginzburg, building on earlier work of Mackey, have defined versions of these indicators which are twisted by an automorphism of the group. In another direction, Linchenko and Montg…
Classically, the exponent of a group is the least common multiple of the orders of its elements. This notion was generalized by Etingof and Gelaki to the context of Hopf algebras. Kashina, Sommerhauser and Zhu later observed that there is a strong connection between exponents and Frobenius-Schur indicators. In this paper, we introduce the notion of twisted exponents and show that there is a similar relationship between the twisted exponent and the twisted Frobenius-Schur indicators defined in previous work of the authors. In particular, we exhibit a new formula for the twisted Frobenius-Schur …
We define total Frobenius-Schur indicator for each object in a spherical fusion category $C$ as a certain canonical sum of its higher indicators. The total indicators are invariants of spherical fusion categories. If $C$ is the representation category of a semisimple quasi-Hopf algebra $H$, we prove that the total indicators are non-negative integers which satisfy a certain divisibility condition. In addition, if $H$ is a Hopf algebra, then all the total indicators are positive. Consequently, the positivity of total indicators is a necessary condition for a quasi-Hopf algebra being gauge equiv…
We generalize the Frobenius-Schur theorem to $C^*$-categories. From this category-theoretical point of view, we introduce the notions of real, complex and quaternionic representations of Hopf $C^*$-algebras. Based on these definitions, we give another type of the Hopf-algebraic analogue of the Frobenius-Schur theorem, originally due to Linchenko and Montgomery. Also given are similar results for weak Hopf $C^*$-algebras, table algebras and compact quantum groups.
We obtain two formulae for the higher Frobenius-Schur indicators: one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist. The first one is a categorical generalization of an analogous result by Kashina, Sommerhauser, and Zhu for Hopf algebras, and the second one extends Bantay's 2nd indicator formula for a conformal field theory to higher degree. These formulae imply the sequence of higher indicators of an object in these categories is periodic. We define the notion of Frobenius-Schur (FS-)exponent of a pivo…
We calculate Frobenius-Schur indicator values for some fusion categories obtained from inclusions of finite groups $H\subset G$, where more concretely $G$ is symmetric or alternating, and $H$ is a symmetric, alternating or cyclic group. Our work is strongly related to earlier results by Kashina-Mason-Montgomery, Jedwab-Montgomery, and Timmer for bismash product Hopf algebras obtained from exact factorizations of groups. We can generalize some of their results, settle some open questions and offer shorter proofs; this already pertains to the Hopf algebra case, while our results also cover fusio…
We study the higher Frobenius-Schur indicators of the representations of the Drinfel'd double of a finite group G, in particular the question as to when all the indicators are integers. This turns out to be an interesting group-theoretic question. We show that many groups have this property, such as alternating and symmetric groups, PSL_2(q), M_{11}, M_{12} and regular nilpotent groups. However we show there is an irregular nilpotent group of order 5^6 with non-integer indicators.
We show that a left-rigid monoidal C^*-category with irreducible monoidal unit is also a sovereign and spherical category. Defining a Frobenius-Schur type indicator we obtain selection rules for the fusion coefficients of irreducible objects. As a main result we prove S_4-invariance of 6j-symbols in such a category.
We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category C, an equivariant indicator of an object in C is defined as a functional on the Grothendieck algebra of the quantum double Z(C) via generalized Frobenius-Schur indicators. The set of all equivariant indicators admits a natural action of the modular group. Using the properties of equivariant indicators, we prove a congruence subgroup theorem for modular categories. As a consequence, all modular representations of a modular category have finite images, and they satisfy a conjecture of Eholz…
Let g be a finite dimensional complex semisimple Lie algebra, and let V be a finite dimensional represenation of g. We give a closed formula for the mth Frobenius-Schur indicator, m>1, of V in representation-theoretic terms. We deduce that the indicators take integer values, and that for a large enough m, the mth indicator of V equals the dimension of the zero weight space of V. For the classical Lie algebras sl(n), so(2n), so(2n+1) and sp(2n), this is the case for m greater or equal to 2n-1, 4n-5, 4n-3 and 2n+1, respectively.
We study the higher Frobenius-Schur indicators of modules over semisimple Hopf algebras, and relate them to other invariants as the exponent, the order, and the index. We prove various divisibility and integrality results for these invariants. In particular, we prove a version of Cauchy's theorem for semisimple Hopf algebras. Furthermore, we give some examples that illustrate the general theory.
We present a new approach to calculating the higher Frobenius-Schur indicators for the simple modules over the Drinfeld double of a finite group. In contrast to the formula by Kashina-Sommerh{ä}user-Zhu that involves a sum over all group elements satisfying a certain condition, our formula operates on the level of conjugacy classes and character tables. It can be implemented in the computer algebra system GAP, efficiently enough to deal, on a laptop, with symmetric groups up to $S_{18}$ (providing further evidence that indicators are non-negative in this case) or simple groups of order up to $…
If G is a finite group and k is a field, there is a natural construction of a Hopf algebra over k associated to G, the Drinfel'd double D(G). We prove that if G is any finite real reflection group with Drinfel'd double D(G) over an algebraically closed field k of characteristic not 2, then every simple D(G)-module has Frobenius-Schur indicator +1. This generalizes the classical results for modules over the group itself. We also prove some new results about Weyl groups. In particular, we prove that any abelian subgroup is inverted by some involution. Also, if E is any elementary abelian 2-subgr…
In this paper, we obtain a canonical central element $ν_H$ for each semi-simple quasi-Hopf algebra $H$ over any field $k$ and prove that $ν_H$ is invariant under gauge transformations. We show that if $k$ is algebraically closed of characteristic zero then for any irreducible representation of $H$ which affords the character $χ$, $χ(ν_H)$ takes only the values 0, 1 or -1, moreover if $H$ is a Hopf algebra or a twisted quantum double of a finite group then $χ(ν_H)$ is the corresponding Frobenius-Schur Indicator. We also prove an analog of a Theorem of Larson-Radford for split semi-simple quasi-…
In this paper we show that for an important class of non-trivial Hopf algebras, the Schur indicator is a computable invariant. The Hopf algebras we consider are all abelian extensions; as a special case, they include the Drinfeld double of a group algebra. In addition to finding a general formula for the indicator, we also study when it is always positive. In particular we prove that the indicator is always positive for the Drinfeld double of the symmetric group, generalizing the classical result for the symmetric group itself.
For M and N finite module categories over a finite tensor category C, the category Rex_C(M,N) of right exact module functors is a finite module category over the Drinfeld center Z(C). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map C-C-bimodule functors to objects of Z(C), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if M and N are exact C-modules and C is pivotal, then the Z(C)-module Rex_C(M,N) is exact. We compute its relative…
We give some applications of a Hopf algebra constructed from a group acting on another Hopf algebra A as Hopf automorphisms, namely Molnar's smash coproduct Hopf algebra. We find connections between the exponent and Frobenius-Schur indicators of a smash coproduct and the twisted exponents and twisted Frobenius-Schur indicators of the original Hopf algebra A. We study the category of modules of the smash coproduct.
We classify pointed fusion categories C(G, $ω$) up to Morita equivalence for 1 < |G| < 32. Among them, the cases |G| = 2 3 , 2 4 and 3 3 are emphasized. Although the equivalence classes of such categories are not distinguished by their Frobenius-Schur indicators, their categorical Morita equivalence classes are distinguished by the set of the indicators and ribbon twists of their Drinfeld centers. In particular, the modular data are a complete invariant for the modular categories Z(C(G, $ω$)) for |G[< 32. We use the computer algebra package GAP and present codes for treating complex-valued gro…
This article investigates duals for bimodule categories over finite tensor categories. We show that finite bimodule categories form a tricategory and discuss the dualities in this tricategory using inner homs. We consider inner-product bimodule categories over pivotal tensor categories with additional structure on the inner homs. Inner-product module categories are related to Frobenius algebras and lead to the notion of $*$-Morita equivalence for pivotal tensor categories. We show that inner-product bimodule categories form a tricategory with two duality operations and an additional pivotal st…
We study the representations and their Frobenius-Schur indicators of two semisimple Hopf algebras related to the symmetric group $S_n$, namely the bismash products $H_n = k^{C_n}# kS_{n-1}$ and its dual $J_n = k^{S_{n-1}}# kC_n = (H_n)^*,$ where $k$ is an algebraically closed field of characteristic 0. Both algebras are constructed using the standard representation of $S_n$ as a factorizable group, that is $S_n = S_{n-1}C_n = C_nS_{n-1}$. We prove that for $H_n$, the indicators of all simple modules are +1. For the dual Hopf algebra $J_n = k^{S_{n_1}}# kC_n$, the indicator can have values eith…
We consider traces on module categories over pivotal fusion categories which are compatible with the module structure. It is shown that such module traces characterise the Morita classes of special haploid symmetric Frobenius algebras. Moreover, they are unique up to a scale factor and they equip the dual category with a pivotal structure. This implies that for each pivotal structure on a fusion category over the complex numbers there exists a conjugate pivotal structure defined by the canonical module trace.
By computing Frobenius-Schur indicators of modules of certain weak Hopf algebras, we give a formula for the number of involutions in symmetric groups, which are contained in a given coset with respect to a given Young subgroup.
We compute higher Frobenius-Schur indicators of pq-dimensional pointed Hopf algebras in characteristic p through their associated graded Hopf algebras. These indicators are gauge invariants for the monoidal categories of representations of these algebras.
In this paper, we define the higher Frobenius-Schur (FS-)indicators for finite-dimensional modules $V$ of a semisimple quasi-Hopf algebra $H$ via the categorical counterpart developed in \cite{NS05}. We prove that this definition of higher FS-indicators coincides with the higher indicators introduced by Kashina, Sommerhäuser, and Zhu when $H$ is a Hopf algebra. We also obtain a sequence of canonical central elements of $H$, which is invariant under gauge transformations, whose values, when evaluated by the character of an $H$-module $V$, are the higher Frobenius-Schur indicators of $V$. As an …
We show the existence of a finite group $G$ having an irreducible character $χ$ with Frobenius-Schur indicator $ν_2(χ){=}{+}1$ such that $χ^2$ has an irreducible constituent $φ$ with $ν_2(φ){=}{-}1$. This provides counterexamples to the positivity conjecture in rational CFT and a conjecture of Zhenghan Wang about pivotal fusion categories.
We present explicit mathematical structures that allow for the reconstruction of the field content of a full local conformal field theory from its boundary fields. Our framework is the one of modular tensor categories, without requiring semisimplicity, and thus covers in particular finite rigid logarithmic conformal field theories. We assume that the boundary data are described by a pivotal module category over the modular tensor category, which ensures that the algebras of boundary fields are Frobenius algebras. Bulk fields and, more generally, defect fields inserted on defect lines, are give…
A modular object in a symmetric monoidal bicategory is a Frobenius algebra object whose product and coproduct are biadjoint, equipped with a braided structure and a compatible twist, satisfying rigidity, ribbon, pivotality, and modularity conditions. We prove that the oriented 3-dimensional bordism bicategory of 1-, 2-, and 3-manifolds is the free symmetric monoidal bicategory on a single anomaly-free modular object.
We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a by-product we obtain results about the Frobenius-Schur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in two-dimensional conformal field theory and present illustrative examples. We show that when the module category is tensor, then it gives ris…
We consider a pivotal monoidal functor whose domain is a modular tensor category (MTC). We show that the trace of such a functor naturally extends to a representation of the corresponding tube category. As irreducible representations of the tube category are indexed by pairs of simple objects in the underlying MTC, the simple multiplicities of this representation form a candidate modular invariant matrix. In general, this matrix will not be modular invariant, however it will always commute with the T-matrix. Furthermore, under certain additional conditions on the original functor, it is shown …
It is well-known that the commutant algebra of the $U_q(\mathfrak{sl}_2)$-action on the $n$-fold tensor product of its fundamental module is isomorphic to the Temperley-Lieb algebra TL$_n(ν)$ with fugacity parameter $ν= -q - q^{-1}$ (at least in the generic case, i.e., when $q$ is not a root of unity, or $n$ is small enough). Furthermore, the simple $U_q(\mathfrak{sl}_2)$-modules appearing in the direct-sum decomposition of the $n$-fold tensor product module are in one-to-one correspondence with those of the Temperley-Lieb algebra. This double-commutant property is referred to as quantum Schur…
These notes offer an introduction to the functorial and algebraic description of 2-dimensional topological quantum field theories `with defects', assuming only superficial familiarity with closed TQFTs in terms of commutative Frobenius algebras. The generalisation of this relation is a construction of pivotal 2-categories from defect TQFTs. We review this construction in detail, flanked by a range of examples. Furthermore we explain how open/closed TQFTs are equivalent to Calabi-Yau categories and the Cardy condition, and how to extract such data from pivotal 2-categories.
We introduce two kinds of gauge invariants for any finite-dimensional Hopf algebra H. When H is semisimple over C, these invariants are respectively, the trace of the map induced by the antipode on the endomorphism ring of a self-dual simple module, and the higher Frobenius-Schur indicators of the regular representation. We further study the values of these higher indicators in the context of complex semisimple quasi-Hopf algebras H. We prove that these indicators are non-negative provided the module category over H is modular, and that for a prime p, the p-th indicator is equal to 1 if, and o…
