Frobenius-Schur Indicators Research Articles

Some quasitensor autoequivalences of Drinfeld doubles of finite groups

We report on two classes of autoequivalences of the category of Yetter–Drinfeld modules over a finite group, or, equivalently the Drinfeld center of the category of representations of a finite group. Both operations are related to the r -th power operation, with r relatively prime to the exponent of the group. One is defined more generally for the group-theoretical fusion category defined by a finite group and an arbitrary subgroup, while the other seems particular to the case of Yetter–Drinfeld modules. Both autoequivalences preserve higher Frobenius–Schur indicators up to Galois conjugation, and they preserve tensor products, although neither of them can in general be endowed with the structure of a monoidal functor.

The pivotal cover and Frobenius–Schur indicators

In this paper, we introduce the notion of the pivotal cover Cpiv of a left rigid monoidal category C to develop a theoretical foundation for the theory of Frobenius–Schur (FS) indicators in “non-pivotal” settings. For an object V∈Cpiv, the (n,r)-th FS indicator νn,r(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 the “adjoint object” in a finite tensor category, which can be considered as a generalization of the adjoint representation of a Hopf algebra. The indicators of this object closely relate to the space of endomorphisms of the iterated tensor product functor.

Characters of graphene’s symmetry group Dg80

We have determined the characters of the diperiodic group Dg80—the symmetry group of the honeycomb lattice. We started from the general formula for construction of irreducible representations (irreps) of symmorphic space groups and have found characters by summing diagonal elements of irrep matrices. The result is given for every element of Dg80, not just for generators. We have calculated the Frobenius–Schur indicator for irreps of Dg80 and proved that all of them are equivalent to real representations.

The real Chevalley involution

AbstractThe Chevalley involution of a connected, reductive algebraic group over an algebraically closed field takes every semisimple element to a conjugate of its inverse, and this involution is unique up to conjugacy. In the case of the reals we prove the existence of a real Chevalley involution, which is defined over$\mathbb{R}$, takes every semisimple element of$G(\mathbb{R})$to a$G(\mathbb{R})$-conjugate of its inverse, and is unique up to conjugacy by$G(\mathbb{R})$. We derive some consequences, including an analysis of groups for which every irreducible representation is self-dual, and a calculation of the Frobenius Schur indicator for such groups.

Generalizations and limitations of string-net models

We ask which topological phases can and can not be realized by exactly soluble string-net models. We answer this question for the simplest class of topological phases, namely, those with Abelian braiding statistics. Specifically, we find that an Abelian topological phase can be realized by a string-net model if and only if (i) it has a vanishing thermal Hall conductance and (ii) it has at least one Lagrangian subgroup, a subset of quasiparticles with particular topological properties. Equivalently, we find that an Abelian topological phase is realizable if and only if it supports a gapped edge. We conjecture that the latter criterion generalizes to the non-Abelian case. We establish these results by systematically constructing all possible Abelian string-net models and analyzing the quasiparticle braiding statistics in each model. We show that the low-energy effective field theories for these models are multicomponent $\text{U}(1)$ Chern-Simons theories, and we derive the $K$-matrix description of each model. An additional feature of this work is that the models we construct are more general than the original string-net models, due to several new ingredients. First, we introduce two new objects $\ensuremath{\gamma},\ensuremath{\alpha}$ into the construction which are related to ${\mathbb{Z}}_{2}$ and ${\mathbb{Z}}_{3}$ Frobenius-Schur indicators. Second, we do not assume parity invariance. As a result, we can realize topological phases that were not accessible to the original construction, including phases that break time-reversal and parity symmetry.

CP violation from finite groups

We discuss the origin of CP violation in settings with a discrete (flavor) symmetry G. We show that physical CP transformations always have to be class-inverting automorphisms of G. This allows us to categorize finite groups into three types: (i) Groups that do not exhibit such an automorphism and, therefore, in generic settings, explicitly violate CP. In settings based on such groups, CP violation can have pure group-theoretic origin and can be related to the complexity of some Clebsch–Gordan coefficients. (ii) Groups for which one can find a CP basis in which all the Clebsch–Gordan coefficients are real. For such groups, imposing CP invariance restricts the phases of coupling coefficients. (iii) Groups that do not admit real Clebsch–Gordan coefficients but possess a class-inverting automorphism that can be used to define a proper (generalized) CP transformation. For such groups, imposing CP invariance can lead to an additional symmetry that forbids certain couplings. We make use of the so-called twisted Frobenius–Schur indicator to distinguish between the three types of discrete groups. With Δ(27), T′, and Σ(72) we present one explicit example for each type of group, thereby illustrating the CP properties of models based on them. We also show that certain operations that have been dubbed generalized CP transformations in the recent literature do not lead to physical CP conservation.

$FSZ$-groups and Frobenius-Schur indicators of quantum doubles

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, PSL2(q), M11, M12 and regular nilpotent groups. However we show there is an irregular nilpotent group of order 5 6 with non-integer indicators.

On some properties of quantum doubles of finite groups

We prove two results about quantum doubles of finite groups over the complex field. The first result is the integrality theorem for higher Frobenius–Schur indicators for wreath product groups SN⋉AN, where A is a finite abelian group. A proof of this result for A=1 appears in a paper by Iovanov, Montgomery, and Mason. The second result is a lower bound for the largest possible number of irreducible representations of the quantum double of a finite group with at most n conjugacy classes. This answers a question asked by Eric Rowell.

How to compute the Frobenius–Schur indicator of a unipotent character of a finite Coxeter system

For each finite, irreducible Coxeter system (W,S), Lusztig has associated a set of “unipotent characters” Uch(W). There is also a notion of a “Fourier transform” on the space of functions Uch(W)→R, due to Lusztig for Weyl groups and to Broué, Lusztig, and Malle in the remaining cases. This paper concerns a certain W-representation ϱW in the vector space generated by the involutions of W. Our main result is to show that the irreducible multiplicities of ϱW are given by the Fourier transform of a unique function ϵ:Uch(W)→{−1,0,1}, which for various reasons serves naturally as a heuristic definition of the Frobenius–Schur indicator on Uch(W). The formula we obtain for ϵ extends prior work of Casselman, Kottwitz, Lusztig, and Vogan addressing the case in which W is a Weyl group. We include in addition a succinct description of the irreducible decomposition of ϱW derived by Kottwitz when (W,S) is classical, and prove that ϱW defines a Gelfand model if and only if (W,S) has type An, H3, or I2(m) with m odd. We show finally that a conjecture of Kottwitz connecting the decomposition of ϱW to the left cells of W holds in all non-crystallographic types, and observe that a weaker form of Kottwitz’s conjecture holds in general. In giving these results, we carefully survey the construction and notable properties of the set Uch(W) and its attached Fourier transform.

Frobenius–Schur indicators of unipotent characters and the twisted involution module

Let W W be a finite Weyl group and σ \sigma a non-trivial graph automorphism of W W . We show a remarkable relation between the σ \sigma -twisted involution module for W W and the Frobenius–Schur indicators of the unipotent characters of a corresponding twisted finite group of Lie type. This extends earlier results of Lusztig and Vogan for the untwisted case and then allows us to state a general result valid for any finite group of Lie type. Inspired by recent work of Marberg, we also formally define Frobenius–Schur indicators for “unipotent characters” of twisted dihedral groups.

Kazhdan–Lusztig cells and the Frobenius–Schur indicator

Let W be a finite Coxeter group. It is well known that the number of involutions in W is equal to the sum of the degrees of the irreducible characters of W. Following a suggestion of Lusztig, we show that this equality is compatible with the decomposition of W into Kazhdan–Lusztig cells. The proof uses a generalisation of the Frobenius–Schur indicator to symmetric algebras, which may be of independent interest.

Modular representations and indicators for bismash products

We introduce Brauer characters for representations of the bismash products of groups in characteristic p>0, p≠2 and study their properties analogous to the classical case of finite groups. We then use our results to extend to bismash products a theorem of Thompson on lifting Frobenius–Schur indicators from characteristic p to characteristic 0.

Frobenius–Schur Indicator for Categories with Duality

We introduce the Frobenius–Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius–Schur theorem including that for semisimple quasi-Hopf algebras, weak Hopf C*-algebras and association schemes. Our framework also clarifies a mechanism of how the “twisted” theory arises from the ordinary case. As a demonstration, we establish twisted versions of the Frobenius–Schur theorem for various algebraic objects. We also give several applications to the quantum SL2.

Strong Reality Properties of Normalizers of Parabolic Subgroups in Finite Coxeter Groups

Let W be a finite Coxeter group, P a parabolic subgroup of W, and N W (P) the normalizer of P in W. We prove that every element in N W (P) is strongly real in N W (P), and that every irreducible complex character of N W (P) has Frobenius-Schur indicator 1.

Twisted Frobenius–Schur indicators for Hopf algebras

The classical Frobenius–Schur indicators for finite groups are character sums defined for any representation and any integer m⩾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 Montgomery have defined Frobenius–Schur indicators for semisimple Hopf algebras. In this paper, the authors construct twisted Frobenius–Schur indicators for semisimple Hopf algebras; these include all of the above indicators as special cases and have similar properties.

Hopf algebras and congruence subgroups

The authors prove that the kernel of the action of the modular group on the centre of a semisimple factorisable Hopf algebra is a congruence subgroup whenever this action is linear. If the action is only projective, they show that the projective kernel is a congruence subgroup. To do this, they introduce a class of generalised Frobenius-Schur indicators and endow it with an action of the modular group that is compatible with the original one.

Semisimple symplectic characters of finite unitary groups

Let G = U ( 2 m , F q 2 ) be the finite unitary group, with q the power of an odd prime p. We prove that the number of irreducible complex characters of G with degree not divisible by p and with Frobenius–Schur indicator −1 is q m − 1 .

Klein Topological Field Theories from Group Representations

We show that any complex (respectively real) representation of finite group naturally generates a open-closed (respectively Klein) topological field theory over complex numbers. We relate the 1-point correlator for the projective plane in this theory with the Frobenius-Schur indicator on the representation. We relate any complex simple Klein TFT to a real division ring.

On the trace of the antipode and higher indicators

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 only if, p is a factor of dimH. As an application, we show the existence of a non-trivial self-dual simple H-module with bounded dimension which is determined by the value of the second indicator.

Frobenius–Schur indicators in Tambara–Yamagami categories

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. We also give some applications to Hopf algebras.