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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Johnson, K.W. (2019). The Extended k-Characters. In: Group Matrices, Group Determinants and Representation Theory. Lecture Notes in Mathematics, vol 2233. Springer, Cham. https://doi.org/10.1007/978-3-030-28300-1_6
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DOI: https://doi.org/10.1007/978-3-030-28300-1_6
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