Character Tables from Class Sums

We extend the notion of conjugacy classes and class sums from finite groups to semisimple Hopf algebras and show that the conjugacy classes are obtained from the factorization of H as irreducible left D(H)-modules. For quasitriangular semisimple Hopf algebras H, we prove that the product of two class sums is an integral combination of the class sums up to d −2 where d = dim H. We show also that in this case the character table is obtained from the S-matrix associated to D(H). Finally, we calculate explicitly the generalized character table of D(kS 3), which is not a character table for any group. It moreover provides an example of a product of two class sums which is not an integral combination of class sums.

The object of this section is the proof of further properties of the class sum correspondence. In particular we study the so-called powermap, i.e the behaviour of corresponding class sums under powers, and collect properties of a finite group determined by its character table. The consequences with respect to the isomorphism problem are the content of the following summarizing result.

In this study, the point groups 𝐷2𝑑 and 𝐶3𝑖 which belong to tetragonal and trigonal crystal systems, respectively, are handled under the class sum approach. Symmetry groups were formed with symmetry elements that left these point groups unchanged and Cayley tables of related groups were obtained. Using these tables, the conjugates of the elements and the classes of the group were formed. Secular equations are written for each class sum obtained by the sum of the elements that make up the class. By solving these secular equations, the character vectors are obtained. Thus, the character tables were reconstructed with the calculated characters for both point groups under the class sum approach.

'. For finite nilpotent groups G and G', and a G-adapted ring S (the rational integers, for example), it is shown that any isomorphism between the centers of the group rings SG and SG' is monomial, i.e., maps class sums in SG to class sums in SG' up to multiplication with roots of unity. As a consequence, G and G' have identical character tables if and only if the centers of their integral group rings ZG and ZG' are isomorphic. In the course of the proof, a new proof of the class sum correspondence is given.

This paper shows how to use information which can be obtained by use of the Todd Coxeter algorithm and related techniques, to reduce a group representation. The method reduces a representation of G given a reduction for a subgroup H of finite index in G. If the character table for G is given, only rational algebraic processes and square roots are used. If the character table for G is not known it can be found by solution of eigenvalue equations, using the class sums, which are obtained in the course of the calculation without scanning the whole group.

When H is a finite dimensional, semisimple, almost cocommutative Hopf algebra, we examine a table of characters which extends the notion of the character table for a finite group. We obtain a formula for the structure constants of the representation ring in terms of values in the character table, and give the example of the quantum double of a finite group. We give a basis of the centre of H which generalizes the conjugacy class sums of a finite group, and express the class equation of H in terms of this basis. We show that the representation ring and the centre of H are dual character algebras (or signed hypergroups).

The class sum operator approach to the representation theory of the point groups O and D4 is described and illustrated by means of several examples. Modified character tables are given for both groups, together with the class multiplication table for O. The construction of tensor operators within the group algebra of each group is discussed, using a modified version of traditional character analysis, and it is found that no E type tensor operator appears in the D4 group algebra.