Markaracter Tables and Q-Conjugacy Character Tables for Cyclic Groups. An Application to Combinatorial Enumeration

Abstract

Abstract Q-Conjugacy character tables for cyclic groups are obtained by starting from character tables. Thus, irreducible representations for a cyclic group are classified into primitive and non-primitive ones. They are collected to form a matrix corresponding to each subgroup. Such a matrix is shown to be a representation (called Q-conjugacy representation) for characterizing Q-conjugacy and dominant classes. The traces of the Q-conjugacy representation are collected to form a Q-conjugacy character table, which is shown to be a square matrix. The elements of such a Q-conjugacy character table for a cyclic group are shown to be integers, which are related to the values of the corresponding character tables. They are also correlated to the markaracter tables for the cyclic group. Characteristic monomial tables for cyclic groups are obtained by starting from the Q-conjugacy character tables and dominant unit-subduced-cycle-index tables. They are applied to combinatorial enumeration of isomers derived from a skeleton belonging to a cyclic group.