TY - CHAP
T1 - Multiplicative Forms on Algebras and the Group Determinant
AU - Johnson, Kenneth W.
N1 - Publisher Copyright:
© Springer Nature Switzerland AG 2019.
PY - 2019
Y1 - 2019
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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U2 - 10.1007/978-3-030-28300-1_1
DO - 10.1007/978-3-030-28300-1_1
M3 - Chapter
AN - SCOPUS:85075170672
T3 - Lecture Notes in Mathematics
SP - 1
EP - 53
BT - Lecture Notes in Mathematics
PB - Springer Verlag
ER -