Group Matrices, Group Determinants and Representation Theory: The Mathematical Legacy of Frobenius

Description

This book sets out an account of the tools which Frobenius used to discover representation theory for nonabelian groups and describes its modern applications. It provides a new viewpoint from which one can examine various aspects of representation theory and areas of application, such as probability theory and harmonic analysis. For example, the focal objects of this book, group matrices, can be thought of as a generalization of the circulant matrices which are behind many important algorithms in information science. The book is designed to appeal to several audiences, primarily mathematicians working either in group representation theory or in areas of mathematics where representation theory is involved. Parts of it may be used to introduce undergraduates to representation theory by studying the appealing pattern structure of group matrices. It is also intended to attract readers who are curious about ideas close to the heart of group representation theory, which do not usually appear in modern accounts, but which offer new perspectives. Front Matter ....Pages i-xxv Multiplicative Forms on Algebras and the Group Determinant (Kenneth W. Johnson)....Pages 1-53 Further Group Matrices and Group Determinants (Kenneth W. Johnson)....Pages 55-110 Norm Forms and Group Determinant Factors (Kenneth W. Johnson)....Pages 111-136 S-Rings, Gelfand Pairs and Association Schemes (Kenneth W. Johnson)....Pages 137-175 The 2-Characters of a Group and the Weak Cayley Table (Kenneth W. Johnson)....Pages 177-209 The Extended k-Characters (Kenneth W. Johnson)....Pages 211-229 Fourier Analysis on Groups, Random Walks and Markov Chains (Kenneth W. Johnson)....Pages 231-270 K-Characters and n-Homomorphisms (Kenneth W. Johnson)....Pages 271-286 Fusion and Supercharacters (Kenneth W. Johnson)....Pages 287-311 Other Situations Involving Group Matrices (Kenneth W. Johnson)....Pages 313-340 Back Matter ....Pages 341-384