# The Extended *k*-Characters

## Abstract

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.

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