### Abstract

The ideas in the initial papers by Frobenius on characters and group determinants are set out and put into context. It is indicated how the theory goes back to the search for “sums of squares identities”, the construction of “hypercomplex numbers” and the investigation of quadratic forms. The underlying objects, the group matrix, and its determinant, the group determinant, are introduced. It is shown that group matrices can be constructed as block circulants. The first construction by Frobenius of group characters for noncommutative groups is explained. This led to his construction of the irreducible factors of the group determinant. The k-characters, used to construct the irreducible factor corresponding to an irreducible character, are defined. Resulting developments are then discussed, with an indication of how the ideas of Frobenius were taken up by other mathematicians and how his approach and its continuation in the theory of norm forms on algebras have been useful. A summary of the various ways in which the work has impacted current areas is included.

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

Publisher | Springer Verlag |

Pages | 1-53 |

Number of pages | 53 |

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. 1-53). (Lecture Notes in Mathematics; Vol. 2233). Springer Verlag. https://doi.org/10.1007/978-3-030-28300-1_1

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*Lecture Notes in Mathematics.*Lecture Notes in Mathematics, vol. 2233, Springer Verlag, pp. 1-53. https://doi.org/10.1007/978-3-030-28300-1_1

**Multiplicative Forms on Algebras and the Group Determinant.** / Johnson, Kenneth W.

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

TY - CHAP

T1 - Multiplicative Forms on Algebras and the Group Determinant

AU - Johnson, Kenneth W.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - The ideas in the initial papers by Frobenius on characters and group determinants are set out and put into context. It is indicated how the theory goes back to the search for “sums of squares identities”, the construction of “hypercomplex numbers” and the investigation of quadratic forms. The underlying objects, the group matrix, and its determinant, the group determinant, are introduced. It is shown that group matrices can be constructed as block circulants. The first construction by Frobenius of group characters for noncommutative groups is explained. This led to his construction of the irreducible factors of the group determinant. The k-characters, used to construct the irreducible factor corresponding to an irreducible character, are defined. Resulting developments are then discussed, with an indication of how the ideas of Frobenius were taken up by other mathematicians and how his approach and its continuation in the theory of norm forms on algebras have been useful. A summary of the various ways in which the work has impacted current areas is included.

AB - The ideas in the initial papers by Frobenius on characters and group determinants are set out and put into context. It is indicated how the theory goes back to the search for “sums of squares identities”, the construction of “hypercomplex numbers” and the investigation of quadratic forms. The underlying objects, the group matrix, and its determinant, the group determinant, are introduced. It is shown that group matrices can be constructed as block circulants. The first construction by Frobenius of group characters for noncommutative groups is explained. This led to his construction of the irreducible factors of the group determinant. The k-characters, used to construct the irreducible factor corresponding to an irreducible character, are defined. Resulting developments are then discussed, with an indication of how the ideas of Frobenius were taken up by other mathematicians and how his approach and its continuation in the theory of norm forms on algebras have been useful. A summary of the various ways in which the work has impacted current areas is included.

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PB - Springer Verlag

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