The Extended k-Characters

The subject of this chapter is the k-characters χ^(k) of a finite group G, and their extensions to more general objects. These characters are constant on certain subsets of G^k, the k-classes. Here work of Vazirani is presented which provides a set of “extended k-characters” for arbitrary k. These connect with various aspects of the representation theory of the symmetric groups and the general linear groups. Immanent k-characters are defined for arbitrary k and any irreducible representation λ of S_n. They coincide with the usual k-characters if λ is the sign character and in the cases k = 2 and k = 3 they had appeared with other names. There are connections with the representation theory of wreath products, with invariant theory and Schur functions. There are orthogonality relations and the Littlewood-Richardson coefficients appear in the decomposition of products of extended k-characters.