The Extended k-Characters

• Published in: Group Matrices, Group Determinants and Representation Theory Vol. 2233; pp. 211 - 229
• Main Author: Johnson, Kenneth W
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2019; Springer International Publishing
• Series: Lecture Notes in Mathematics
• ISBN: 3030282996; 9783030282998
• ISSN: 0075-8434; 1617-9692
• DOI: 10.1007/978-3-030-28300-1_6

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 Gk, 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 Sn. 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.