Generalized circulants and class functions of finite groups

The regular representation of the ring of class functions of a finite group is used to construct families of matrices that enjoy many of the properties of circulant matrices. For different choices of finite groups one gets different families of matrices: the circulants, block circulants, block circulants with circulant blocks of all levels, and many others. The properties of these families are simple consequences of the character theory of finite groups. Many known properties of circulants and some of their generalizations are obtained as special cases of the properties of the families constructed. Some applications to character theory are also mentioned.