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On Higher Frobenius-Schur Indicators

Yevgenia Kashina       Yorck Sommerhäuser Yongchang Zhu

Abstract

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.

Introduction

For a finite group, one can evaluate a character on the sum of all m-th powers of the group elements. The resulting number, divided by the order of the group, is called the m-th Frobenius-Schur indicator of the character. The first use of these indicators was made by F. G. Frobenius and I. Schur (cf. ) to give a criterion when a representation of a finite group can be realized by matrices with real entries - for this question, it is the second indicator that is relevant. This is also meaningful for other fields than the complex numbers: Here the indicator tells whether or not a given module is self-dual.

Higher indicators, i.e., indicators with m > 2, arise when one considers the root number function in a finite group. This function assigns to a group element the number of its m-th roots, i.e., the number of group elements whose m-th power is equal to the given element. It is clear that this number depends only on the conjugacy class, and therefore defines a class function that can be expanded in terms of the irreducible characters. Using the orthogonality relations for characters, it is not hard to see that the coefficient of an irreducible character in this expansion is its m-th Frobenius-Schur indicator (cf. , Lem. (4.4), p. 49).

For Hopf algebras, Frobenius-Schur indicators were first considered by V. Linchenko and S. Montgomery on the one hand (cf. ) and by J. Fuchs, A. Ch. Ganchev, K. Szlachányi, and P. Vecsernyés on the other hand (cf. ). Here, the sum of the m-th powers of the group elements is replaced by the m-th Sweedler power of the integral. The authors then use the indicators, or at least the second indicator, to prove an analogue of the criterion of Frobenius and Schur whether or not a representation is self-dual: The Frobenius-Schur theorem asserts that this depends on whether the second indicator is 0, 1, or -1.

The topic of the present writing are the higher Frobenius-Schur indicators for semisimple Hopf algebras and their relation to other invariants of irreducible characters. These other invariants are the order, the multiplicity, the exponent, and the index. Let us briefly describe the nature of these invariants. The notion of the order of an irreducible character is a generalization of the notion of the order of an element in a finite group: It is the smallest integer such that the corresponding tensor power contains a nonzero invariant subspace. The dimension of this invariant subspace is called the multiplicity of the irreducible character. An irreducible character has order 1 if and only if it is trivial, and has order 2 if and only if it is self-dual. In these cases, the multiplicity of the character is 1.

The exponent of a semisimple Hopf algebra is another invariant, which generalizes the exponent of a group (cf. ). The exponent of a module is a slight generalization of this concept: In the group case, it is the exponent of the image of the group in the representation. There are various ways to generalize this concept to semisimple Hopf algebras; we will use Sweedler powers on the one hand and a certain canonical tensor on the other hand.

The next invariant that we study, the index of imprimitivity, arises from Perron-Frobenius theory. It is clear that the matrix representation of the left multiplication by the character of a module with respect to the basis consisting of all irreducible characters has nonnegative integer entries. As we will explain below, the corresponding Perron-Frobenius eigenvalue, i.e., the positive eigenvalue that has the largest possible absolute value, is the degree of the character. However, since the entries of the above matrix are in general not strictly positive, this eigenvalue is not necessarily strictly greater than the absolute values of the other characters, so that there can be other eigenvalues which are not positive, but have the same absolute value. As we will see, in the most interesting cases the above matrix is indecomposable; in this case, the number of such eigenvalues is called the index of imprimitivity.

The text is organized as follows: In Section 1, we discuss the formalism of Sweedler powers in an arbitrary bialgebra. A Sweedler power of an element in a bialgebra is constructed by applying the comultiplication several times, permuting the arising tensor factors and multiplying them together afterwards. This notion is a slight modification of the original notion (cf. ), where the tensor factors were not permuted, and has the advantage that iterated Sweedler powers are still Sweedler powers. We then consider the Sweedler powers that arise from a certain special kind of permutations. This is motivated by the fact that the values of characters on the Sweedler powers of the integral lie in certain cyclotomic fields, and this special kind of Sweedler powers is well adapted to describe the action of the Galois group on these values.

From Section 2 on, we consider semisimple Hopf algebras over algebraically closed fields of characteristic zero. We prove a first formula for the higher Frobenius-Schur indicators that should be understood as a generalization of the Frobenius-Schur theorem for these indicators-in particular, it implies the Frobenius-Schur theorem for the second indicators. This first formula describes Frobenius-Schur indicators in terms of a certain operator on the corresponding tensor power of the module, and we establish several other properties of this operator as well.

In Section 3, we then consider the exponent of a module and prove a second formula for the higher Frobenius-Schur indicators that uses a certain canonical tensor. Combining this with the first formula, we prove a version of Cauchy's theorem for Hopf algebras: A prime that divides the dimension of a semisimple Hopf algebra must also divide its exponent. This result was conjectured by P. Etingof and S. Gelaki (cf. ); it was known in the case of the prime 2 (cf. ). Furthermore, we prove that the higher indicators are integers if the exponent is squarefree.

In Section 4, we define the notion of the order and the multiplicity of a module and prove that the order of a module divides its multiplicity times the dimension of the Hopf algebra. This result generalizes the theorem that a semisimple Hopf algebra that has a nontrivial self-dual simple module must have even dimension (cf. ) to modules of arbitrary orders - nontrivial self-dual simple modules are of order 2, and the multiplicity is 1 in this case. In particular, we get in this way a fully independent new proof of the old theorem.

In Section 5, we study the index of imprimitivity, or briefly the index, of the matrix that represents the left multiplication by a character with respect to the canonical basis that we have in the character ring - the basis consisting of the irreducible characters. The main result of this section is a precise formula for the index in terms of central grouplike elements. Essentially, the result says that the eigenvalues of the above matrix that have the same absolute value as the degree are obtained by evaluating the character at certain central grouplike elements. As a consequence of this formula, we see that the index divides the order as well as the exponent.

In Section 6, we apply a new tool - the Drinfel'd double of the Hopf algebra. We prove that, by restricting modules over the Drinfel'd double to the Hopf algebra, we get a map from the character ring of the Drinfel'd double onto the center of the character ring of the Hopf algebra. From this, we deduce that the center of the rational character ring of the Hopf algebra, i.e., the span of the irreducible characters over the rational numbers, is isomorphic to a product of subfields of the cyclotomic field whose order is the exponent of the Hopf algebra. Finally, we deduce a third formula for the Frobenius-Schur indicators in terms of the action of the Drinfel'd element on the induced module over the Drinfel'd double.

In Section 7, we finally compute explicitly a number of examples. In this way, we can limit the possible generalizations of the results that we have obtained. The class of examples that we study are certain extensions of group rings by dual group rings. In particular, the Drinfel'd doubles of finite groups belong to this class.

Throughout the whole exposition, we consider a base field that is denoted by K. All vector spaces considered are defined over K, and all tensor products without subscripts are taken over K. Unless stated otherwise, a module is a left module. The set of natural numbers is the set N:={1,2,3,...}; in particular, 0 is not a natural number. The symbol Qn denotes the n-th cyclotomic field, and not the field of n-adic numbers, and Zn denotes the set Z/nZ of integers modulo n, and not the ring of n-adic integers.

From Section 2 on, we assume that the base field K is algebraically closed of characteristic zero. H denotes a semisimple Hopf algebra with coproduct Δ, counit ε, and antipode S. We will use the same symbols to denote the corresponding structure elements of the dual Hopf algebra H*. Note that a semisimple Hopf algebra is automatically finite-dimensional (cf. , Chap. V, Exerc. 4, p. 108). By results of R. G. Larson and D. E. Radford (cf. , Thm. 3.3, p. 276; , Thm. 4, p. 195), H is also cosemisimple and involutory, i.e., H* is semisimple and the antipode is an involution. From Maschke's theorem (cf. , Thm. 2.2.1, p. 20), we get that there is a unique two-sided integral Λ ∈ H such that ε(Λ)=1; this element will be used heavily throughout.

Furthermore, we use the convention that propositions, definitions, and similar items are referenced by the paragraph in which they occur; they are only numbered separately if this reference is ambiguous.