Multiplicative Forms on Algebras and the Group Determinant

Research output: Chapter in Book/Report/Conference proceedingChapter

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 languageEnglish (US)
Title of host publicationLecture Notes in Mathematics
PublisherSpringer Verlag
Pages1-53
Number of pages53
DOIs
StatePublished - Jan 1 2019

Publication series

NameLecture Notes in Mathematics
Volume2233
ISSN (Print)0075-8434
ISSN (Electronic)1617-9692

Fingerprint

Multiplicative
Determinant
Algebra
Frobenius
Matrix Groups
Irreducible Character
Sum of squares
Quadratic form
Continuation
Form
Norm
Character

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

Johnson, K. W. (2019). Multiplicative Forms on Algebras and the Group Determinant. In 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
Johnson, Kenneth W. / Multiplicative Forms on Algebras and the Group Determinant. Lecture Notes in Mathematics. Springer Verlag, 2019. pp. 1-53 (Lecture Notes in Mathematics).
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Johnson, KW 2019, Multiplicative Forms on Algebras and the Group Determinant. in 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.

Lecture Notes in Mathematics. Springer Verlag, 2019. p. 1-53 (Lecture Notes in Mathematics; Vol. 2233).

Research output: Chapter in Book/Report/Conference proceedingChapter

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Johnson KW. Multiplicative Forms on Algebras and the Group Determinant. In Lecture Notes in Mathematics. Springer Verlag. 2019. p. 1-53. (Lecture Notes in Mathematics). https://doi.org/10.1007/978-3-030-28300-1_1