### 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.

Original language | English (US) |
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Title of host publication | Lecture Notes in Mathematics |

Publisher | Springer Verlag |

Pages | 211-229 |

Number of pages | 19 |

DOIs | |

State | Published - Jan 1 2019 |

### Publication series

Name | Lecture Notes in Mathematics |
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Volume | 2233 |

ISSN (Print) | 0075-8434 |

ISSN (Electronic) | 1617-9692 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Lecture Notes in Mathematics*(pp. 211-229). (Lecture Notes in Mathematics; Vol. 2233). Springer Verlag. https://doi.org/10.1007/978-3-030-28300-1_6

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*Lecture Notes in Mathematics.*Lecture Notes in Mathematics, vol. 2233, Springer Verlag, pp. 211-229. https://doi.org/10.1007/978-3-030-28300-1_6

**The Extended k-Characters.** / Johnson, Kenneth W.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - The Extended k-Characters

AU - Johnson, Kenneth W.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

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U2 - 10.1007/978-3-030-28300-1_6

DO - 10.1007/978-3-030-28300-1_6

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T3 - Lecture Notes in Mathematics

SP - 211

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PB - Springer Verlag

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